Global continua of solutions to the Lugiato–Lefever model for frequency combs obtained by two-mode pumping

Abstract

We consider Kerr frequency combs in a dual-pumped microresonator as time-periodic and spatially $2\pi$-periodic traveling wave solutions of a variant of the Lugiato–Lefever equation, which is a damped, detuned and driven nonlinear Schrödinger equation given by $\textrm{i}a_\tau =(\zeta -\textrm{i})a-d a_{x x}-|a|^2a+\textrm{i}f_0+\textrm{i}f_1\textrm{e}^{\textrm{i}(k_1 x-\nu _1 \tau )}$. The main new feature of the problem is the specific form of the source term $f_0+f_1\textrm{e}^{\textrm{i}(k_1 x-\nu _1 \tau )}$ which describes the simultaneous pumping of two different modes with mode indices $k_0=0$ and $k_1\in \mathbb {N}$. We prove existence and uniqueness theorems for these traveling waves based on a-priori bounds and fixed point theorems. Moreover, by using the implicit function theorem and bifurcation theory, we show how non-degenerate solutions from the 1-mode case, i.e., $f_1=0$, can be continued into the range $f_1\not =0$. Our analytical findings apply both for anomalous ($d>0$) and normal ($d<0$) dispersion, and they are illustrated by numerical simulations.

1 Introduction

Optical frequency comb devices are extremely promising in many applications such as, e.g., optical frequency metrology [30], spectroscopy [25, 32], ultrafast optical ranging [29] and high capacity optical communications [19]. For many of these applications, the Kerr soliton combs¹ are generated by using a monochromatic pump. However, recently new pump schemes have been discussed, where more than one resonator mode is pumped, cf. [28]. The pumping of two modes can have a number of important advantages. In particular, 1-solitons arising from a dual-pump scheme can be spectrally broader and spatially more localized than 1-solitons arising from a monochromatic pump, cf. [10] for a comprehensive discussion of the theoretical advantages. Mathematically, Kerr comb dynamics are described by the Lugiato–Lefever equation (LLE), a damped, driven and detuned nonlinear Schrödinger equation [12, 17, 21]. Our analysis relies on a variant of the LLE which is modified for two-mode pumping, cf. [28] and [10] for a derivation. Using dimensionless, normalized quantities this equation takes the form

$$\begin{aligned} \textrm{i}a_\tau =(\zeta -\textrm{i})a-d a_{x x}-|a|^2a+\textrm{i}f_0+\textrm{i}f_1\textrm{e}^{\textrm{i}(k_1 x-\nu _1 \tau )}, \qquad a \ 2\pi \text {-periodic in } x. \end{aligned}$$

(1)

Here, $a(\tau ,x)$ represents the optical intracavity field as a function of normalized time $\tau =\frac{\kappa }{2}t$ and angular position $x\in [0,2\pi ]$ within the ring resonator. The constant $\kappa >0$ describes the cavity decay rate and $d=\frac{2}{\kappa }d_2$ quantifies the dispersion in the system (where $\omega _k = \omega _0+d_1k+d_2k^2$ is the cavity dispersion relation between the resonant frequencies $\omega _k$ and the relative indices $k\in \mathbb {Z}$). Here, the case $d<0$ amounts to normal and the case $d>0$ to anomalous dispersion. The resonant modes in the cavity are numbered by $k\in \mathbb {Z}$ with $k_0=0$ being the first and $k_1\in \mathbb {N}$ the second pumped mode. With $f_0,f_1$, we describe the normalized power of the two input pumps and $\omega _{p_0},\omega _{p_1}$ denote the frequencies of the two pumps. Since there are now two pumped modes, there are also two normalized detuning parameters denoted by $\zeta =\frac{2}{\kappa }(\omega _0-\omega _{p_0})$ and $\zeta _1=\frac{2}{\kappa }(\omega _{k_1}-\omega _{p_1})$. They describe the offsets of the input pump frequencies $\omega _{p_0}$ and $\omega _{p_1}$ to the closest resonance frequency $\omega _0$ and $\omega _{k_1}$ of the microresonator. The particular form of the pump term $\textrm{i}f_0+\textrm{i}f_1\textrm{e}^{\textrm{i}(k_1x-\nu _1 \tau )}$ with $\nu _1=\zeta -\zeta _1+d k_1^2$ suggests to change into a moving coordinate frame and to study solutions of (1) of the form $a(\tau ,x)=u(s)$ with $s=x-\omega \tau$ and $\omega =\frac{\nu _1}{k_1}$. These traveling wave solutions propagate with speed $\omega$ in the resonator and their profiles u solve the ordinary differential equation

$$\begin{aligned} -d u''+\textrm{i} \omega u'+(\zeta -\textrm{i})u-|u|^2 u+\textrm{i}f_0+\textrm{i}f_1 \textrm{e}^{\textrm{i}k_1 s}=0, \qquad u \ 2\pi \text {-periodic}. \end{aligned}$$

(2)

In the case $f_1=0$, equation (1) amounts to the case of pumping only one mode. This case has been thoroughly studied. For example, the method of spatial dynamics and center manifold reduction has been utilized in [11, 12, 20] to prove existence and bifurcation of stationary solutions. Here, the $2\pi$-periodicity is relaxed and a detailed normal form analysis of the real four-dimensional first-order system corresponding to (2) is performed to track periodic, quasiperiodic and homoclinic orbits near the curve of constant solutions. A similar approach has been taken in [21,22,23] for anomalous, normal and even third-order dispersion, where numerical continuation methods revealed snaking behavior of bifurcation curves as well as solitary structures far from the curve of trivial solutions, cf. also [24] for the prediction of periodicity changes along bifurcation curves. A different point of view based on the $2\pi$-periodic boundary conditions and the bifurcation theory of Crandall–Rabinowitz was taken in [8, 18] where additionally global bifurcation pictures allowed parameter studies concerning quality measures such as comb bandwidth and power conversion efficiency. Both points of view (spatial dynamics, boundary value problem) have been used in [9, 20] for (2) on the entire real line shadowing the results for localized solutions on finite intervals.

Also dynamical questions for (2) in the case $f_1=0$ have been considered. Using again bifurcation theory and spatial dynamics, the stability or instability of spatially periodic stationary solutions with respect to co-periodic or subharmonic perturbations is shown in [6, 7, 13]. Alternatively, assuming spectral stability with a simple zero eigenvalue and the rest of the spectrum bounded away from the imaginary axis the authors of [27] showed asymptotic stability based on the Gearhart–Greiner–Prüss Theorem. The first result allowing to transfer spectral stability to nonlinear stability with respect to just localized perturbations for stationary solutions on the real line appeared in [14].

In this paper, we are interested in the case $f_1\ne 0$. Since the specific form of the forcing term is not essential for many of our results, we allow in the following for more general forcing terms

$$\begin{aligned} f(s)=f_0+ f_1 e(s) \end{aligned}$$

with a $2\pi$-periodic (not necessarily continuous) function $e:\mathbb {R}\rightarrow \mathbb {C}$ and $f_0, f_1\in \mathbb {R}$. Hence, we consider the LLE

$$\begin{aligned} -d u''+\textrm{i} \omega u'+(\zeta -\textrm{i})u-|u|^2 u+\textrm{i}f(s)=0, \qquad u \ 2\pi \text {-periodic}. \end{aligned}$$

(3)

Our main results on the existence of solutions to (3) are stated in Sect. 2. In Sect. 3, we illustrate our main analytical results by numerical simulations. The proofs of the main results are given in Sect. 4 (a-priori bounds), Sect. 5 (existence and uniqueness), and Sect. 6 (continuation results). The appendix contains a technical result and a consideration of the case where in (2) the value $k_1$ is not an integer but close to an integer. Let us finally mention that our existence results are concerned with $2\pi$-periodic solutions. We do not claim any solitary character. Finding solitary solutions is another task that has been pursued in the recent work [10] from an applied point of view.

2 Main results

In the following, we state our main results. We distinguish between trivial and non-trivial solutions of (3). The former means a constant solution and the latter a non-constant solution. Trivial (constant) solutions of (3) only exist in the case when $f_1=0$ or e is constant. Moreover, we also distinguish between results on one-sided and two-sided continuation. The distinction arises since in some results the information on continua² of solution pairs $(f_1,u)$ can only be given for $f_1\ge 0$ or $f_1 \le 0$ (one-sided continuation), whereas in other results (requiring further assumptions) we can provide information on continua of solution pairs $(f_1,u)$ without sign restriction on $f_1$ (two-sided continuation).

Existence and uniqueness: Theorem 1 provides existence of at least one solution of (3) for any choice of the parameters and any choice of f. Theorem 7 in Sect. 5 is a corresponding uniqueness result, which applies whenever $|\zeta |\gg 1$ is sufficiently large or (essentially) $\Vert f\Vert _2\ll 1$ is sufficiently small.

Continuation of trivial solutions: Theorem 2 and Corollary 1 describe how trivial (constant) solutions from the special case $f_1=0$ can be continued into non-trivial solutions for $f_1\not =0$.

Continuation of non-trivial solutions: Theorem 3 and Corollary 2 show how a non-trivial solution from the case $f_1=0$ can be continued to $f_1\not =0$.

Two-sided continuations: The previous continuation results were one-sided. In Sect. 2.3, we exploit underlying symmetries of the forcing term and provide results on two-sided continuations. In particular, in Theorem 4 we provide additional information on the local shape of the two-sided continuation of trivial solutions.

We will use the following Sobolev spaces. For $k\in \mathbb {N}$, the space $H^k(0,2\pi )$ consists of all square-integrable functions on $(0,2\pi )$ whose weak derivatives up to order k exist and are square-integrable on $(0,2\pi )$. By $H^k_{\text {per}}(0,2\pi )$, we denote all locally square-integrable $2\pi$-periodic functions on $\mathbb {R}$ whose weak derivatives up to order k exist and are locally square-integrable on $\mathbb {R}$. In both spaces, the norm is given by $\Vert u\Vert = \bigl (\sum _{j=0}^k \Vert (\frac{d}{ds})^j u\Vert _{L^2(0,2\pi )}^2 \bigr )^{1/2}$. Clearly $H^k_{\text {per}}(0,2\pi )$ is a proper subspace of $H^k(0,2\pi )$ since $u\in H^k_{\text {per}}(0,2\pi )$ implies that $(\frac{d}{ds})^{j}u(0)=(\frac{d}{ds})^{j}u(2\pi )$ for $j=0,\ldots ,k-1$. Unless otherwise stated, all of the above Hilbert spaces are spaces of complex-valued functions over the field $\mathbb {R}$. In particular, for $v,w\in L^2(0,2\pi )$ we use the inner product $\langle v,w\rangle _2{:}{=}{\text {Re}}\int _0^{2\pi } v{\overline{w}}\,ds$. The induced norm is denoted by $\Vert \cdot \Vert _{2}$.

Our first theorem, which ensures the existence of a solution of (3) in the general case where $f_1$ does not need to vanish, is based on a-priori bounds and a variant of Schauder’s fixed point theorem known as Schaefer’s fixed point theorem, cf. Theorem 6. A corresponding uniqueness result based on Banach’s contraction mapping theorem is given in Theorem 7 in Sect. 5.

Theorem 1

Equation (3) has at least one solution $u\in H^2_{\text {per}}(0,2\pi )$ for any choice of the parameters $d\in \mathbb {R}{\setminus }\{0\}$, $\zeta ,\omega \in \mathbb {R}$ and any choice of $f\in H^2(0,2\pi )$.

Next we address the question whether a known solution $u_0$ of (3) for $f_1=0$ can be continued into the regime $f_1\not =0$. This continuation will be done differently depending on whether $u_0$ is constant (trivial) or non-constant (non-trivial). Moreover, we first concentrate on one-sided continuations for $f_1>0$ (or $f_1<0$). Two-sided continuations will be discussed in Sect. 2.3.

2.1 One-sided continuation of trivial solutions

In the special case $f_1=0$, there are trivial (constant) solutions $u_0 \in \mathbb {C}$ of (3) satisfying the algebraic equation

$$\begin{aligned} (\zeta -\textrm{i})u_0-|u_0|^2 u_0+\textrm{i}f_0=0. \end{aligned}$$

(4)

From [18, Lemma 2.1], we know that for given $f_0 \in \mathbb {R}$ the curve of constant solutions can be parameterized by

$$\begin{aligned} \zeta (t)=(1-t^2)f_0^2+\frac{t}{\sqrt{1-t^2}}, \quad u_0(t)=(1-t^2)f_0-\text {i} f_0 t \sqrt{1-t^2}, \quad t\in (-1,1). \end{aligned}$$

(5)

In Fig. 1, we show the curve of the squared $L^2$-norm of all constant solutions of (3) for $f_1=0$ and $f_0=1$, $f_0=\frac{2\sqrt{2}}{\root 4 \of {27}}$ and $f_0=2$. The curve may or may not have turning points which are characterized by $\zeta '(t)=0$. This condition can be formulated independently of t by the equivalent condition $\zeta ^2-4|u_0|^2 \zeta +1+3 |u_0|^4=0$.

Fig. 1

Curve of squared $L^2$-norm of all constant solutions of (3) for $f_1=0$ and $f_0=1$ (green), $f_0=\frac{2\sqrt{2}}{\root 4 \of {27}}$ (red) and $f_0=2$ (blue) when $\zeta \in [-1,5]$. Turning points (if they exist) are marked with a cross

By a straightforward analysis, one can show that with $f^*=\frac{2\sqrt{2}}{\root 4 \of {27}}$, we have

• no turning point for $|f_0|<f^*$ (cf. Fig. 1 green curve),

• exactly one (degenerate) turning point for $|f_0|=f^*$ (cf. Fig. 1 red curve),

• exactly two turning points for $|f_0|>f^*$ (cf. Fig. 1 blue curve).

Note that for $|f_0|>f^*$, as a consequence of the existence of two turning points, three different constant solutions exist for certain values of $\zeta$.

Starting from $f_1=0$, we use a kind of global implicit function theorem (cf. Theorem 8) to continue a constant solution $u_0 \in \mathbb {C}$ of (3) with respect to $f_1$. This procedure is analyzed in Theorem 2. The continuation works if the constant solution $u_0\in \mathbb {C}$ is non-degenerate in the following sense.

Definition 1

A solution $u\in H^2_{\text {per}}(0,2\pi )$ of (3) for $f_1=0$ is called non-degenerate if the kernel of the linearized operator

$$\begin{aligned} L_u\varphi {:}{=}-d \varphi '' +\textrm{i}\omega \varphi ' +(\zeta -\textrm{i}-2|u|^2)\varphi -u^2\overline{\varphi }, \quad \varphi \in H^2_{\text {per}}(0,2\pi ) \end{aligned}$$

consists only of ${{\,\textrm{span}\,}}\{u'\}$.

Remark 1

Note that $L_u: H^2_{\text {per}}(0,2\pi ) \rightarrow L^2(0,2\pi )$ is a compact perturbation of the isomorphism $-d\frac{d^2}{dx^2}+{{\,\textrm{sign}\,}}(d): H^2_{\text {per}}(0,2\pi ) \rightarrow L^2(0,2\pi )$ and hence an index-zero Fredholm operator. Notice also that ${{\,\textrm{span}\,}}\{u'\}$ always belongs to the kernel of $L_u$ as can be seen from differentiating (3) for $f_1=0$ w.r.t. s. Non-degeneracy means that except for the obvious candidate $u'$ (and its real multiples), there is no other element of the kernel of $L_u$. Notice also that a solution $u_0$ is non-degenerate if the linearized operator $L_{u_0}$ is injective, and, as a consequence, invertible in suitable spaces.

Non-degeneracy of trivial solutions is discussed in the next lemma and the subsequent remark. Non-degeneracy of non-trivial solutions has been proven for a particular class of solutions in certain parameter regimes in [7].

Lemma 1

A trivial solution $u_0\in \mathbb {C}$ of (3) for $f_1=0$ is non-degenerate if and only if

• (a) Case $\omega \not =0$:

  $$\begin{aligned} \zeta ^2-4|u_0|^2 \zeta +1+3 |u_0|^4 \ne 0. \end{aligned}$$

• (b) Case $\omega =0$:

  $$\begin{aligned} (\zeta +d m^2)^2-4|u_0|^2(\zeta +d m^2)+1+3|u_0|^4\ne 0 \quad \text { for all } m\in \mathbb {N}_0. \end{aligned}$$

Proof

Let $\varphi \in H^2_{\text {per}}(0,2\pi )$ be in the kernel of the linearized operator, i.e.,

$$\begin{aligned} -d \varphi '' +\textrm{i}\omega \varphi ' +(\zeta -\textrm{i}-2|u_0|^2)\varphi -u_0^2\overline{\varphi }=0. \end{aligned}$$

This implies that the Fourier coefficients $\varphi _m$ of the Fourier series $\varphi =\sum _{m\in \mathbb {Z}} \varphi _m \textrm{e}^{\textrm{i}ms}$ have the property that

$$\begin{aligned} (d m^2-\omega m+\zeta -\textrm{i}-2|u_0|^2)\varphi _m -u_0^2 \overline{\varphi _{-m}}=0 \end{aligned}$$

for all $m\in \mathbb {Z}$. If we also write down the complex conjugate of this equation

$$\begin{aligned} -\overline{u_0}^2 \varphi _{m}+(d m^2+\omega m+\zeta +\textrm{i}-2|u_0|^2)\overline{\varphi _{-m}}=0 \end{aligned}$$

then we see that non-degeneracy of $u_0$ is equivalent to the nonvanishing of the determinant for this two-by-two system in the variables $\varphi _m, \overline{\varphi _{-m}}$ for all $m\in \mathbb {N}_0$. Computing the determinant, we obtain the condition

$$\begin{aligned} \begin{aligned} (\zeta +d m^2)^2-4|u_0|^2(\zeta +dm^2)+1+3|u_0|^4-\omega ^2 m^2 -2\text {i} \omega m\ne 0 \quad \text{ for } \text{ all } m\in \mathbb {N}_0. \end{aligned} \end{aligned}$$

(6)

In the case $\omega \ne 0$, this is trivially satisfied for all $m\ne 0$ (because then the imaginary part is nonzero) and for $m=0$ by assumption (a) of the lemma. In the case $\omega =0$, condition (6) can only be guaranteed by assumption (b). $\square$

Remark 2

Trivial solutions of (3) for $f_1=0$ are determined by (4). For $\omega \not =0$, all trivial solutions $u_0$ of (3) for $f_1=0$ are non-degenerate except those at the turning points described above. In the case $\omega =0$, all trivial solutions $u_0$ of (3) for $f_1=0$ are non-degenerate except those at the (potential) bifurcation points and the turning points. This is true (up to additional conditions ensuring transversality and simplicity of kernels) because the necessary condition for bifurcation w.r.t. $\zeta$ from the curve of trivial solutions is fulfilled if and only if the expression in (b) vanishes for at least one $m\in \mathbb {N}$, cf. [8, 18].

Theorem 2

Let $d \in \mathbb {R}{\setminus }\{0\}$, $\zeta ,\omega ,f_0\in \mathbb {R}$ and $e \in H^2(0,2\pi )$ be fixed. Let furthermore $u_0 \in \mathbb {C}$ be a constant non-degenerate solution of (3) for $f_1=0$. Then, the maximal continuum³$\mathcal {C}^+\subset [0,\infty ) \times H^2_{\text {per}}(0,2\pi )$ of solutions $(f_1,u)$ of (3) with $(0,u_0)\in {\mathcal {C}}^+$ has the following properties:

• (i) locally near $(0,u_0)$ the set ${\mathcal {C}}^+$ is the graph of a smooth curve $f_1\mapsto (f_1,u(f_1))$,

• (ii) ${\mathcal {C}}^+\cap [0,M]\times H^2_{\text {per}}(0,2\pi )$ is bounded for any $M>0$.

Moreover, if ${\text {pr}}_1({\mathcal {C}}^+)$ denotes the projection of ${\mathcal {C}}^+$ onto the $f_1$-parameter component, then at least one of the following properties hold:

• (a) ${\text {pr}}_1({\mathcal {C}}^+)= [0,\infty )$,

or

• (b) $\exists u_0^+ \not = u_0: \, (0,u_0^+) \in {\mathcal {C}}^+.$

A maximal continuum ${\mathcal {C}}^-\subset (-\infty ,0]\times H^2_{\text {per}}(0,2\pi )$ with corresponding properties also exists.

Remark 3

If property (a) of Theorem 2 holds, then ${\mathcal {C}}^+$ is unbounded in the direction of the parameter $f_1\in [0,\infty )$, and hence, this is an existence result for all $f_1 \in [0,\infty )$. Property (b) means that the continuum ${\mathcal {C}}^+$ returns to the $f_1=0$ line at a point $u_0^+ \not = u_0$.

Corollary 1

Property (a) in Theorem 2 holds in any of the following three cases,

• (i) $\displaystyle {{\,\textrm{sign}\,}}(d)\zeta<-C(d,f_0)^2 {\textbf{1}}_{d<0}-27\biggl (1+\frac{\pi f_0^2 |\omega |}{|d|}+\frac{\pi ^2 f_0^4}{|d|}\biggr ) C(d,f_0)^6$,

• (ii) $\displaystyle {{\,\textrm{sign}\,}}(d)\zeta >3C(d,f_0)^2+\frac{\omega ^2}{4|d|}$,

• (iii) $\displaystyle \sqrt{3} C(d,f_0)<1$,

where

$$\begin{aligned} C(d,f_0) =|f_0|(1+2\pi ^2 f_0^2 |d|^{-1}). \end{aligned}$$

In particular $|\zeta | \gg 1$ or $|f_0| \ll 1$ is sufficient.

2.2 One-sided continuation of non-trivial solutions

One can ask the question whether also non-trivial (non-constant) solutions at $f_1=0$ may be continued into the regime of $f_1>0$. This depends on two issues: existence and non-degeneracy of a non-trivial solution of (3) for $f_1=0$. First we note that for $\omega =0$, there is a plethora of non-trivial solutions, cf. [8, 18]. For $\omega \not =0$, we do not know whether non-trivial solutions exist for $f_1=0$. Note that for $\omega \not =0$, there are no bifurcations from the curve of trivial solutions, since bifurcation requires degeneracy of trivial solutions other than the turning points. However, this is impossible by Lemma 1(a). Moreover, the recent paper [3] shows that the continuation of non-trivial solutions for $\omega =0$ into the regime $\omega \not =0$ typically fails. This strongly indicates (but does not prove) that for $\omega \not =0$ there may be no solutions other than the trivial ones. Although by the current state of understanding the hypotheses of Theorem 3 (see below) can only be fulfilled for $\omega =0$, we allow in the following for general $\omega \in \mathbb {R}$.

In order to describe the continuation from a non-degenerate non-trivial solution, let us first state some properties of (3) for $f_1=0$: if $u_0$ solves (3) for $f_1=0$ and if we denote its shifts by $u_\sigma (s){:}{=}u_0(s-\sigma )$, then $u_\sigma$ also solves (3) for $f_1=0$. Hence,

$$\begin{aligned} S:\left\{ \begin{array}{rcl} \mathbb {R}&{} \rightarrow &{} \mathbb {R}\times H^2_\text {per}(0,2\pi ), \\ \sigma &{} \mapsto &{} (0,u_{\sigma }) \end{array} \right. \end{aligned}$$

describes a trivial curve of solutions of (3) from which we wish to bifurcate at some point $(0,u_{\sigma _0})$. The bifurcation will be achieved by applying the Crandall–Rabinowitz theorem on bifurcation from a simple eigenvalue, cf. Theorem 9, with $\sigma$ being the bifurcation parameter. The bifurcating local curve will be reparametrized with $f_1$ as a new parameter and then globally continued by the already introduced global version of the implicit function theorem, cf. Theorem 8.

Recall also from non-degeneracy that $\ker L_{u_{\sigma }}={{\,\textrm{span}\,}}\{u_\sigma '\}$. Since $L_{u_\sigma }^*$ also has an one-dimensional kernel, there exists $\phi _\sigma ^*\in H^2_\text {per}(0,2\pi )$ such that $\ker L_{u_\sigma }^*={{\,\textrm{span}\,}}\{\phi _\sigma ^*\}$. Notice that $\phi _\sigma ^*(s) = \phi _0^*(s-\sigma )$. Finally, $\sigma _0$ will be determined in such a way that there exists a unique solution $\xi _{\sigma _0}\in H^2_\text {per}(0,2\pi )$ of

$$\begin{aligned} L_{u_0}\xi _{\sigma _0} =-\textrm{i}e(\cdot +\sigma _0) \end{aligned}$$

with the property that $\xi _{\sigma _0}\perp _{L^2} u_0'$. Details of the construction of $\sigma _0$ and $\xi _{\sigma _0}$ will be given in Lemma 2.

Theorem 3

Let $d \in \mathbb {R}{\setminus }\{0\}$, $\zeta ,\omega ,f_0\in \mathbb {R}$ and $e \in H^2(0,2\pi )$ be fixed. Let furthermore $u_0\in H^2_{\text {per}}(0,2\pi )$ be a non-trivial non-degenerate solution of (3) for $f_1=0$. If $\sigma _0\in \mathbb {R}$ satisfies

$$\begin{aligned} {\text {Im}}\int \limits _0^{2\pi }e(s+\sigma _0)\overline{\phi _0^*(s)}\,ds =0 \end{aligned}$$

(7)

and

$$\begin{aligned} {\text {Im}}\int \limits _0^{2\pi }e'(s+\sigma _0)\overline{\phi _0^*(s)}\,ds \ne 0 \end{aligned}$$

(8)

then the maximal continuum ${\mathcal {C}}^+\subset [0,\infty ) \times H^2_{\text {per}}(0,2\pi )$ of solutions $(f_1,u)$ of (3) with $(0,u_0)\in {\mathcal {C}}^+$ has the following properties:

• (i) there exists a smooth curve $C:[0,\delta ) \rightarrow {\mathcal {C}}^+$ with $C(t)=(f_1(t), u(t))$, $\dot{f_1}(0)=1$, $C(0)=(0,u_{\sigma _0})$ such that locally near $(0,u_{\sigma _0})$ all solutions $(f_1,u)$ of (3) with $f_1\ge 0$ lie on the curve S or on the curve C,

• (ii) ${\mathcal {C}}^+\cap [0,M]\times H^2_{\text {per}}(0,2\pi )$ is bounded for any $M>0$.

Moreover, if zero is an algebraically simple eigenvalue of $L_{u_0}$ then there exists a connected set ${\mathcal {C}}^{+}_* \subset {\mathcal {C}}^+$ with ${\text {pr}}_1({\mathcal {C}}^{+}_*)\subset (0,\infty )$ and $(0,u_{\sigma _0})\in \overline{{\mathcal {C}}^{+}_*}$ which satisfies at least one of the following properties:

• (a) ${\text {pr}}_1({\mathcal {C}}^+_*)=(0,\infty )$,

or

• (b) $\exists u_0^+ \not = u_{\sigma _0}: \, (0, u_0^+) \in \overline{{\mathcal {C}}^{+}_*}$.

A maximal continuum ${\mathcal {C}}^-\subset (-\infty ,0]\times H^2_{\text {per}}(0,2\pi )$ with corresponding properties also exists.

Remark 4

($\alpha$) It follows from the implicit function theorem that (7) is a necessary condition for bifurcation (non-trivial kernel of the linearization). Assumption (8) amounts to the transversality condition. Both (7) and (8) together mean that the map $\sigma \mapsto {\text {Im}}\int _0^{2\pi }e(s+\sigma )\overline{\phi _0^*(s)}\,ds$ has a simple zero $\sigma _0$. This map generically has $2k_1$ simple sign-changes if we assume $e(s+\frac{\pi }{k_1})=-e(s)$ for some $k_1\in \mathbb {N}$ as for the prototypical case $e(s)=\textrm{e}^{\textrm{i}k_1 s}$. More on the determination of $\sigma _0$ in the prototypical case can be found in the subsequent Corollary 2 and Remark 5.

($\beta$) Note that in property (b), we exclude that $u_0^+=u_{\sigma _0}$ but we do not exclude that $u_0^+$ coincides with a shift of $u_0$ different from $u_{\sigma _0}$.

For the special choice $e(s)=\textrm{e}^{\textrm{i}k_1 s}$, Theorem 3 takes the following form.

Corollary 2

Let $k_1 \in \mathbb {N}$, $e(s)=\textrm{e}^{\textrm{i}k_1 s}$ and $d,\zeta ,\omega ,f_0,u_0$ be as in Theorem 3. Assume that

$$\begin{aligned} \int _0^{2\pi } \textrm{e}^{\textrm{i}k_1 s}\overline{\phi _0^*(s)}\,ds \ne 0 \end{aligned}$$

(9)

and that $\sigma _0\in \mathbb {R}$ satisfies

$$\begin{aligned} \tan (k_1\sigma _0) = \frac{\int _0^{2\pi }\cos (k_1s){\text {Im}}\phi _0^*(s)-\sin (k_1 s){\text {Re}}\phi _0^*(s)\,ds}{\int _0^{2\pi }\sin (k_1s){\text {Im}}\phi _0^*(s)+\cos (k_1 s){\text {Re}}\phi _0^*(s)\,ds}. \end{aligned}$$

(10)

Then, the conditions (7) and (8) of Theorem 3 hold.

Remark 5

($\alpha$) If (9) is satisfied then assumption (10) is a necessary condition for bifurcation.

($\beta$) Assumption (9) in Corollary 2 guarantees that the numerator and the denominator of the right-hand side of (10) do not vanish simultaneously. In the case where the denominator vanishes, Equation (10) is to be read as $\cos (k_1\sigma _0)=0$. In the interval $[0,\tfrac{\pi }{k_1})$ equation (10) has a unique solution $\sigma _0\in [0,\tfrac{\pi }{k_1})$. All solutions of (10) in $[0,2\pi )$ are then given by $\sigma _0+j\tfrac{\pi }{k_1}$ for $j=0,\ldots ,2k_1-1$. This can result in up to $2 k_1$ bifurcation points. Smaller periodicities of $u_0$ may reduce the actual number of different bifurcation points. For example, if $k_1\ge 2$ and if $u_0$ has smallest period $\tfrac{2\pi }{k_1}$ then only two bifurcation points exist.

($\gamma$) Let $j\in \mathbb {N}$ not be a divisor of $k_1$ and $u_0$ be $\tfrac{2\pi }{j}$-periodic. Then, assumption (9) is not satisfied since $\phi _0^*$ inherits the periodicity of $u_0$. We will say more about this case in the Appendix.

($\delta$) The non-trivial solutions $u_0$ of (3) for $f_1=0$ and $\omega =0$ constructed in [8, 18] are even around $s=0$. For such $u_0$, the value of $\sigma _0$ in Corollary 2 is determined by the simpler expression

$$\begin{aligned} \tan (k_1\sigma _0) = -\frac{\int _0^{2\pi } \sin (k_1 s){\text {Re}}\phi _0^*(s)\,ds}{\int _0^{2\pi }\sin (k_1s){\text {Im}}\phi _0^*(s)\,ds}. \end{aligned}$$

It is an open problem if (3) admits solutions for $f_1=0$ and $\omega =0$ which (up to a shift) are not even around $s=0$.

2.3 Two-sided continuations

Here, we explain how we can use the results of Theorems 2 and 3, as well as Corollary 2 for the continua ${\mathcal {C}}^+$ and ${\mathcal {C}}^-$ in order to obtain two-sided continua w.r.t. the parameter component $f_1$.

As a first trivial observation, we can construct a two-sided continuum in the following way both for the setting of Theorems 2 and 3: let ${\mathcal {C}}\subset \mathbb {R}\times H^2_{\text {per}}(0,2\pi )$ be the maximal continuum of solutions $(f_1,u)$ of (3) with $(0,u_0)\in {\mathcal {C}}$. Then, ${\mathcal {C}}$ contains both ${\mathcal {C}}^+$ and $\mathcal {C^-}$.

Next we assume that the generalized forcing term $f(s)=f_0+f_1 e(s)$ satisfies the symmetry condition that $e\bigl (s+\frac{\pi }{k_1}\bigr )=-e(s)$ for some $k_1\in \mathbb {N}$. This symmetry condition is motivated by (2) where $e(s)=\textrm{e}^{\textrm{i}k_1 s}$. If we denote by R the reflection operator which acts on solution pairs and is given by

$$\begin{aligned} R: (f_1,u) \mapsto \bigl (-f_1, u\bigl (\cdot +\tfrac{\pi }{k_1}\bigr )\bigr ) \end{aligned}$$

then, again both for the setting of Theorems 2 and 3, the continuum ${\mathcal {C}}$ has the following property:

$$\begin{aligned} (f_1,u)\in {\mathcal {C}} \Leftrightarrow R(f_1,u)\in {\mathcal {C}}. \end{aligned}$$

This shows that globally the solution sets for positive and negative $f_1$ only differ by a phase shift. The following global structure result is a consequence of this symmetry.

Proposition 1

Let $d \in \mathbb {R}{\setminus }\{0\}$, $\zeta ,\omega ,f_0\in \mathbb {R}$ and $e\in H^2(0,2\pi )$ be such that $e\bigl (s+\frac{\pi }{k_1}\bigr )=-e(s)$ for some $k_1 \in \mathbb {N}$. Let furthermore $u_0$ be a solution of (3) for $f_1=0$. Then, the maximal continua ${\mathcal {C}}^+$, ${\mathcal {C}}^-$ and ${\mathcal {C}}$ containing $(0,u_0)$ satisfy $\mathcal {C^-}=R({\mathcal {C}}^+)$ and ${\mathcal {C}}\supset {\mathcal {C}}^+ \cup {\mathcal {C}}^-.$

Proof

It is obvious that ${\mathcal {C}}\supset {\mathcal {C}}^+ \cup {\mathcal {C}}^-$. Now we prove that $\mathcal {C^-}=R({\mathcal {C}}^+)$. Clearly, ${\mathcal {C}}^+$ and $R({\mathcal {C}}^+)$ contain all shifts $\{(0,u_{\sigma }):\sigma \in \mathbb {R}\}$. Since additionally $R({\mathcal {C}}^+) \subset (-\infty ,0]\times H^2_{\text {per}}(0,2\pi )$ is connected we find that $R({\mathcal {C}}^+) \subset {\mathcal {C}}^-$. If we assume that $R({\mathcal {C}}^+) \subsetneq {\mathcal {C}}^-$ then we obtain ${\mathcal {C}}^+ \subsetneq R^{-1}({\mathcal {C}}^-)$, which contradicts the maximality of ${\mathcal {C}}^+$. $\square$

As another consequence, we have that either ${\text {pr}}_1({\mathcal {C}})=(-\infty ,\infty )$ or ${\text {pr}}_1({\mathcal {C}})$ is bounded from above and below. In the latter case, we call ${\mathcal {C}}$ a loop.

Our final result builds upon Theorem 2 and the resulting two-sided continuation of a trivial solution $u_0$. It describes the shape of the $L^2$-projection of the continuum ${\mathcal {C}}$ locally near $(0,u_0)$. In particular, local convexity or concavity can be read from this result. In Sect. 3, we will put this result into perspective with numerical simulations of the $f_1$-continuation of trivial solutions.

Theorem 4

Assume that the assumptions of Theorem 2 are satisfied and that additionally $e(s)=\textrm{e}^{\textrm{i}k_1 s}$ is fixed for a $k_1 \in \mathbb {N}$. Then, we can determine the local shape of the curve $f_1\mapsto \Vert u(f_1)\Vert _2^2$ as follows:

$$\begin{aligned} \frac{d}{df_1} \Vert u(f_1)\Vert _2^2\mid _{f_1=0} = 0, \qquad \frac{d^2}{df_1^2} \Vert u(f_1)\Vert _2^2\mid _{f_1=0}= 4\pi ({\text {Re}}(u_0 \overline{\epsilon })+|\alpha |^2+|\beta |^2) \end{aligned}$$

with

$$\begin{aligned} \alpha&= \frac{-\textrm{i}(dk_1^2+k_1\omega +\zeta +\textrm{i}-2|u_0|^2)}{(\zeta +dk_1^2-2|u_0|^2)^2-(\omega k_1+\textrm{i})^2-|u_0|^4},\\ \beta&= \frac{\textrm{i} u_0^2}{(\zeta +dk_1^2-2|u_0|^2)^2-(\omega k_1-\textrm{i})^2-|u_0|^4},\\ x&=\zeta -\textrm{i}-2|u_0|^2,\\ y&=-u_0^2,\\ z&= 4u_0(|\alpha |^2+|\beta |^2)+4\overline{u_0}\alpha \beta ,\\ \epsilon&= \frac{-{\overline{z}} y+z{\overline{x}}}{|x|^2-|y|^2}. \end{aligned}$$

3 Numerical illustration of the analytical results

In this section, we restrict ourselves to equation (2), i.e., we fix $e(s)=\textrm{e}^{\textrm{i}k_1 s}$. For this choice, we know from Sect. 2.3 that the one-sided continua ${\mathcal {C}}^+$ and ${\mathcal {C}}^-$ are related by ${\mathcal {C}}^-= R({\mathcal {C}}^+)$. The following numerical examples were computed with $d=-0.1$, $f_0=2$, $k_1=1$ and $\omega =1$.

Fig. 2

Continua of solutions $(f_1,u)$ of (2) for selected values of the detuning $\zeta$. The other parameters were set to $d=-0.1$, $f_0=2$, $k_1=1$, and $\omega =1$

Figure 2 illustrates some of the two-sided continua ${\mathcal {C}}^+ \cup {\mathcal {C}}^-$ obtained by continuation of trivial solutions for different values of the detuning $\zeta$. Every point on the black and colored curves corresponds to a solution u of (2), but for the sake of visualization in a three-dimensional image every solution has to be represented by a single number. In Fig. 2, the quantity $\tfrac{1}{2\pi }\Vert u\Vert _2^2$ was used for this purpose.

The black curve corresponds to spatially constant solutions of (2) obtained for $f_1=0$ and $\zeta \in [2.4,4.3]$. The colored curves represent (parts of) the continua associated to these solutions. Every trivial solution (possibly except the ones at turning points) has an associated continuum, but for the sake of visualization these continua are only shown for selected values of $\zeta$, namely $\zeta \in \{2.4, 2.6, \ldots , 4.0, 4.2\}$. The picture is symmetric with symmetry plane $\{(\zeta ,0,z) :\zeta \in \mathbb {R}, z\in \mathbb {R}\}$. This is an immediate consequence of the relation ${\mathcal {C}}^-= R({\mathcal {C}}^+)$ and the fact that shifting u does not change $\Vert u\Vert _2.$

For $\zeta \in \{2.4, \, 2.6, \, 4.2\}$, there is only one trivial solution, and for these three values Fig. 2 shows a part of the associated two-sided continuum ${\mathcal {C}}^+ \cup {\mathcal {C}}^-$. Although $f_1$ was restricted to $[-2,2]$, each of these continua appears to be global in $f_1$, i.e., we conjecture that the continua continue for all values $f_1\in (-\infty ,\infty )$. This corresponds to case (a) in Theorem 2.

For $\zeta \in \{2.8, \, 3.0, \ldots , \, 4.0\}$, however, there are three trivial solutions. For these values of $\zeta$, there is one colored loop which connects two solutions, and one continuum which seems to continue for all values of $f_1$. The former corresponds to case (b) in Theorem 2, the latter to case (a). For $\zeta \in \{2.8, \, 3.0\}$, the “lower” two solutions are connected, whereas for $\zeta \in \{3.2, \, \ldots , \, 4.0\}$ it is the “upper” two solutions which are connected. Hence, there seems to be a threshold value $\zeta ^*$ that determines which of the two scenarios occurs. Computations with more values of $\zeta$ show that this threshold value $\zeta ^*$ lies between 3.1344 and 3.1359; cf. Fig. 3. The union of the continua for $\zeta$-values close to the threshold $\zeta ^*$ (i.e., for $\zeta =3.1344$ and $\zeta =3.1359$) is nearly the same, and the two continua nearly meet in two points.⁴ The mathematical mechanisms which cause this qualitative change are not yet understood. One could expect that the connectivity threshold coincides with the value where the square of the $L^2$-norm of the solutions as a function of $f_1$ changes from being locally convex to locally concave. However, Theorem 4 shows that this is not true.

Fig. 3

Same situation as in Fig. 2. Zoom to the region close to the threshold where the continua change connectivity

Fig. 4

Same situation as in Fig. 2, but depicted from a different angle and with more values of $\zeta$

Figure 4 illustrates the same application, but depicted from a different angle and with more values of $\zeta$. Repeating the simulation with $d=0.1$ (anomalous dispersion) instead of $d=-0.1$ (normal dispersion) did not change the picture essentially. Figures 2, 3 and 4 were generated by discretizing (2) with central finite differences (1000 grid points), and by applying the classical continuation method as described in, e.g., [1], to the discretized system.

The result of Theorem 4 can be interpreted as follows: each point on the trivial curve is a local extremum of the squared $L^2$-norm of the solution curve $f_1\mapsto u(f_1)$. The type of local extremum is described by the sign of the second derivative $\frac{d^2}{df_1^2} \Vert u(f_1)\Vert _2^2\mid _{f_1=0}$. We visualize this by an example for $d=-0.1$, $f_0=2$, $k_1=1$, $\omega =1$. By using the parameterization $t\mapsto \zeta (t), t\mapsto u_0(t)$ for $t\in (-1,1)$ from (5) we can illustrate the sign-changes of the second derivative. In Fig. 5, we are plotting the curve $t \mapsto (\zeta (t), |u_0(t)|^2)$ and indicate at each point on the curve the sign of $4\pi ({\text {Re}}(u_0(t) \bar{\epsilon }(t))+|\alpha (t)|^2+|\beta (t)|^2)$, where $\epsilon (t), \alpha (t), \beta (t)$ are taken from Theorem 4 with $\zeta =\zeta (t)$ and $u_0=u_0(t)$.

Fig. 5

Sign of the second derivative of $f_1\mapsto \Vert u(f_1)\Vert _2^2$ at $f_1=0$; blue$=$positive, red$=$negative

In this particular example, as we run through the curve of trivial solutions from left to right a first sign-change of $\frac{d^2}{df_1^2} \Vert u(f_1)\Vert _2^2\mid _{f_1=0}$ occurs at $\zeta \approx 0.8533$. A second sign-change (in fact a singularity changing from $-\infty$ to $+\infty )$ occurs at the first turning point. Then, the next sign-change occurs on the part of the branch between the two turning points at $\zeta \approx 3.34$. Finally, the second turning point generates the last sign-change from $-\infty$ to $+\infty$. Clearly, the changes in the nature of the local extremum of $f_1\mapsto \Vert u(f_1)\Vert _2^2$ at $f_1=0$ do not correspond to the topology changes of the solution continua which occur near the threshold value $\zeta ^*\in (3.1344,3.1359)$.

Next, we keep the parameters $d=-0.1$, $f_0=2$, $k_1=1$ but choose $\omega =0$ instead of $\omega =1$. Recall that for $\omega =0$ there is a plethora of non-trivial solutions of (2) for $f_1=0$, cf. [8, 18]. In fact, this time we find additional primary and secondary bifurcation branches for $f_1=0$ which are illustrated in Fig. 6 in gray and brown, respectively. Bifurcation points are shown as gray dots. The bifurcation branches consist of non-trivial solutions. Further, some numerical approximations of the two-sided maximal continua ${\mathcal {C}}$ obtained by continuation of trivial or non-trivial solutions for different values of the detuning $\zeta$ are shown. If we start from a constant solution at $f_1=0$, then ${\mathcal {C}}^\pm$ are described by Theorem 2. Likewise, if we start from a non-constant solution at $f_1=0$ which has no smaller period than $2\pi$, then ${\mathcal {C}}^\pm$ are described by Theorem 3. In both cases, ${\mathcal {C}}\supset {\mathcal {C}}^+ \cup {\mathcal {C}}^-$ by Proposition 1, but in all examples below we observe in fact equality. If we expect a maximal continuum to contain two or more (non-trivial) different simple closed curves, then we illustrate the latter ones with different colors. Let us look at some particular values of $\zeta$ where different phenomena occur.

Fig. 6

Continua of solutions $(f_1,u)$ of (2) for selected values of the detuning $\zeta$. The other parameters were set to $d=-0.1$, $f_0=2$, $k_1=1$, and $\omega =0$

At $\zeta =2.7$, we see exactly one solution for $f_1=0$. This solution is constant and its continuation appears to be global in $f_1$. For $\zeta =3.9$ and $f_1=0$, we see three constant solutions but also one non-constant solution (up to shifts) which lies on one of the gray bifurcation branches. The continuation of the constant solution with smallest magnitude again appears to be global in $f_1$, while the other three solutions lie on the same eight-shaped maximal continuum which we will denote as figure eight continuum. Note that the latter continuum contains all shifts of the non-trivial solution for $f_1=0$.

The figure eight can be interpreted as an outcome of Theorem 2 applied to one of the constant solutions on the figure eight. Here, case (b) of the theorem applies. However, the figure eight can also be interpreted as an outcome of Theorem 3 applied to the non-constant solution $u_0$ at $f_1=0$. Again, case (b) of the theorem applies. A plot (which we omit) of the non-trivial solution $u_0$ at $f_1=0$ shows that $u_0$ has no smaller period than $2 \pi$. Thus, according to Remark 5.($\beta$) exactly two shifts of it, which differ by $\pi$, are bifurcation points. To sum up, we observe that the figure eight continuum in fact contains a simple closed figure eight curve which exactly goes through two shifts of $u_0$ (which differ by $\pi$) in the point where the orange lines intersect the gray line of non-trivial solutions. The two shifts cannot be distinguished in the picture, because a shift does not change the $L^2$-norm.

Fig. 7

Zoom at $\zeta =3.6$

To illustrate the different continua for $\zeta =3.6$, we provide a zoom in Fig. 7. We obtain again an unbounded continuum and a figure eight continuum. However, here we also find a third maximal continuum which cannot be found by simply continuing one of the constant solutions. This continuum consists of the blue and the light blue simple closed curve connected to each other by shifts at $f_1=0$. The parts of the blue and the light blue curve in the region $f_1\ge 0$ are described by case (b) of Theorem 3 applied to one of the non-trivial solutions $u_0$ at $f_1=0$ on it. They have no smaller period than $2\pi$ (plots not shown). Going from the blue part to the light blue part is a consequence of reflection. At $f_1=0$, the blue curve intersects the gray line at exactly two points. The light blue curve does the same, but at $\pi$-shifts of these points.

For $\zeta =3.3$, the situation is more complicated, see the zoom in Fig. 8. In this case, we see three constant solutions for $f_1=0$ but also seven non-constant ones. The continuation of the upper constant solution (orange) appears to be unbounded. We observe that the blue, the red and the green simple closed curve in fact form a single maximal continuum, since all curves are connected by shifts of non-constant solutions at $f_1=0$. Viewed from top to bottom, we find (plots not shown) that the first, the third and the last one are $\pi$-periodic while the remaining ones have smallest period $2\pi$. All together, we observe that exactly two shifts of every non-constant solution at $f_1=0$ are bifurcation points. For the solutions which have no smaller period than $2\pi$, this is a direct consequence of Theorem 3, cf. Remark 5.($\beta$). However, at the three remaining $\pi$-periodic solutions at $f_1=0$ Theorem 3 does not apply, cf. Remark 5.($\gamma$). Nevertheless, we observe continuations from these points. Interestingly, these points seem to be characterized by horizontal tangents, at least in this example.

Fig. 8

Zoom at $\zeta =3.3$

For $\zeta =3$, we see three constant solutions and four non-constant ones at $f_1=0$. Again, the continuation of the upper constant solution is unbounded. We provide a more general investigation in Fig. 9, where we also depict several of the continued solutions u of (2) for $f_1\ne 0$. Since u is complex valued, we use the quantity $|u(s)|^2$ for illustration purposes and plot it against $s\in [-\pi ,\pi ]$. In Fig. 9a, we show a bounded continuum consisting of the light blue and the red simple closed curve connected to each other by shifts at $f_1=0$. Starting from the constant solution on the light blue curve and proceeding first into the $f_1>0$ direction, Fig. 9b, c shows plots of functions corresponding to colored triangles. In Fig. 9d–f, functions corresponding to colored dots on the red curve are shown, where we start again at the constant solution and initially proceed in the $f_1>0$ direction. We observe that both curves cross the ($\pi$-periodic) non-constant solution with second largest norm, but at two different shifts: the leftmost dark-red curves in (c) and (f) only coincide after a nonzero shift. Continuations from $\pi$-periodic solutions at $f_1=0$ are not covered by Theorem 3. Nevertheless, they are observed in the numerical experiments, again with horizontal tangents. The explanation of these continuations remains open, cf. the Appendix for further discussion.

Fig. 9

Zoom at $\zeta =3$ and illustration of selected functions

4 Proof of a-priori bounds

We use the notation $r_+=\max \{0,r\}$ to denote the positive part of any real number $r\in \mathbb {R}$ and also ${\textbf{1}}_{d<0}$ to denote (as a function of $d\in \mathbb {R}$) the characteristic function of the interval $(-\infty ,0)$. We write $\Vert \cdot \Vert _p$ for the standard norm on $L^p(0,2\pi )$ for $p \in [1,\infty ]$. A continuous map between two Banach spaces is said to be compact if it maps bounded sets into relatively compact sets.

Theorem 5

Let $d \in \mathbb {R}{\setminus }\{0\}$, $\zeta ,\omega \in \mathbb {R}$ and $f \in H^2(0,2\pi )$. Then for every solution $u \in H^2_{\text {per}}(0,2\pi )$ of (3), the a-priori bounds

$$\begin{aligned} \Vert u\Vert _2&\le F, \end{aligned}$$

(11)

$$\begin{aligned} \Vert u'\Vert _2&\le B\Vert u\Vert _2^{\frac{1}{4}}\le B F^{\frac{1}{4}} , \end{aligned}$$

(12)

$$\begin{aligned} \Vert u\Vert _\infty&\le C \end{aligned}$$

(13)

hold, where

$$\begin{aligned} F&= F(f)= \Vert f\Vert _2, \\ B&= B(d,f)=\frac{F^{\frac{11}{4}}}{2|d|}+2\Vert f'\Vert _{\infty } F^{\frac{1}{4}}+\sqrt{\Vert f''\Vert _2 F^{\frac{1}{2}}+2\Vert f'\Vert _{\infty }\bigg (\sqrt{\frac{F}{2\pi }} +1\bigg )}, \\ C&= C(d,f)=\frac{F}{\sqrt{2\pi }}+\sqrt{2\pi } B F^{\frac{1}{4}}. \end{aligned}$$

For $\zeta {{\,\textrm{sign}\,}}(d)\ll -C^2 {\textbf{1}}_{d<0}$, these bounds can be improved to

$$\begin{aligned} \Vert u\Vert _2 \le D, \quad \Vert u\Vert _\infty \le \biggl (\frac{F^{\frac{3}{4}}}{\sqrt{2\pi }}+\sqrt{2\pi }B\biggr )D^{\frac{1}{4}}, \end{aligned}$$

where

$$\begin{aligned} D=D(d,f,\omega ,\zeta )=\biggl (\frac{F^{\frac{3}{2}}+ |\omega |B F^{\frac{3}{4}} +|d|B^2}{(-\zeta {\text {sign}}(d)-C^2 {\textbf{1}}_{d<0})_+}\biggr )^{\frac{2}{3}}. \end{aligned}$$

Remark 6

The improvement in the second part of the theorem lies in the fact that the bound D becomes small when the detuning $\zeta$ is such that $\zeta {{\,\textrm{sign}\,}}(d)$ is very negative.

Proof

The proof is divided into five steps.

Step 1. We first prove the $L^2$ estimate

$$\begin{aligned} \Vert u\Vert _2 \le F= \Vert f\Vert _2. \end{aligned}$$

(14)

To this end we multiply the differential equation (3) with ${\bar{u}}$ to obtain

$$\begin{aligned} -d u'' {\bar{u}}+\textrm{i} \omega u' \bar{u}+(\zeta -\textrm{i})|u|^2-|u|^4+\textrm{i}f {\bar{u}}=0. \end{aligned}$$

(15)

Taking the imaginary part yields

$$\begin{aligned} -d {\text {Im}}(u'' {\bar{u}})+\omega {\text {Re}}(u'{\bar{u}})-|u|^2+{\text {Re}}(f {\bar{u}})=0. \end{aligned}$$

(16)

Let $h{:}{=}|u|^2-{\text {Re}}(f {\bar{u}})$, $H{:}{=}-d{\text {Im}}(u'\bar{u})+\frac{\omega }{2}|u|^2$. Then, $H'=h$ by equation (16) and $H(0)=H(2\pi )$ by the periodicity of u. Hence,

$$\begin{aligned} 0=H(2\pi )-H(0)=\int _0^{2 \pi } \, h \,ds = \int _0^{2 \pi } \, |u|^2-{\text {Re}}(f {\bar{u}}) \,ds \end{aligned}$$

which implies

$$\begin{aligned} \Vert u\Vert _2^2=\int _0^{2\pi } \, {\text {Re}}(f {\bar{u}}) \, ds \le \Vert f\Vert _2 \Vert u\Vert _2 = F\Vert u\Vert _2. \end{aligned}$$

Step 2. Next we prove

$$\begin{aligned} \Vert u'\Vert _2 \le B\Vert u\Vert _2^{\frac{1}{4}}\le B F^{\frac{1}{4}}. \end{aligned}$$

(17)

From (3), we may isolate the linear term u and insert its derivative $u'$ into the following calculation for $\Vert u'\Vert _2^2$:

$$\begin{aligned} \begin{aligned} \Vert u'\Vert _2^2=&{} {\text{ Re }}\int _0^{2\pi }\, u' {\bar{u}}' \, ds {\mathop {=}\limits ^{(3)}}{\text{ Re }}\int _0^{2\pi } \, (\text {i} d u''+\omega u'-\text {i}\zeta u+\text {i}|u|^2 u + f)' {\bar{u}}' \, ds \\=&{} {\text{ Re }}\int _0^{2\pi } \, \text {i} d u'''{\bar{u}}'+\omega u'' {\bar{u}}' -\text {i}\zeta |u'|^2+\text {i}(|u|^2 u)'{\bar{u}}' + f'{\bar{u}}' \, ds \\=&{} \int _0^{2\pi } \, -d({\text{ Im }}(u''{\bar{u}}'))'+\left( \frac{\omega }{2}|u'|^2\right) ' \, ds - {\text{ Im }}\int \limits _0^{2\pi } \, (|u|^2 u)'{\bar{u}}' \, ds +{\text{ Re }}\int _0^{2\pi } \, f'{\bar{u}}' \, ds \\=&{} \int _0^{2\pi } \, (|u|^2)' {\text{ Im }}({\bar{u}} u') -{\text{ Re }}(f'' {\bar{u}}) \, ds +{\text{ Re }}f' {\bar{u}} \big \vert _0^{2\pi } \\\le&{} \int _0^{2\pi } \, \frac{1}{d}(|u|^2)' \Big (\frac{\omega }{2}|u|^2-H\Big )ds +\Vert f''\Vert _2 \Vert u\Vert _2 +2\Vert f'\Vert _{\infty } \Vert u\Vert _{\infty } \\=&{} \int _0^{2\pi } \, \frac{\omega }{4d} (|u|^4)'-\frac{1}{d}(|u|^2)' H \, ds+\Vert f''\Vert _2 \Vert u\Vert _2 +2\Vert f'\Vert _{\infty } \Vert u\Vert _{\infty } \\=&{} \int _0^{2\pi } \, -\frac{1}{d}(|u|^2)'(H-H(0)) ds +\Vert f''\Vert _2 \Vert u\Vert _2 +2\Vert f'\Vert _{\infty } \Vert u\Vert _{\infty }. \end{aligned} \end{aligned}$$

Next notice the pointwise estimate

$$\begin{aligned} h =|u|^2-{\text {Re}}(f {\bar{u}}) \ge |u|^2-|f||u| \ge - \frac{1}{4} |f|^2 \end{aligned}$$

from which we deduce the following two-sided estimate for $H-H(0)$:

$$\begin{aligned} H(s)-H(0)&=\int _0^s \, h(r) \, dr \ge -\frac{1}{4}\Vert f\Vert _2^2 \quad (s\in [0,2\pi ]) \quad \text { and } \\ H(s)-H(0)&=H(s)-H(2\pi )=-\int _s^{2\pi } \, h(r) \, dr \le \frac{1}{4}\Vert f\Vert _2^2 \quad (s\in [0,2\pi ]). \end{aligned}$$

Continuing the above inequality for $\Vert u'\Vert _2^2$, we conclude

$$\begin{aligned} \Vert u'\Vert _2^2 \le \frac{ \Vert f\Vert _2^2}{2|d|} \Vert u\Vert _2 \Vert u'\Vert _2 +\Vert f''\Vert _2 \Vert u\Vert _2 +2\Vert f'\Vert _{\infty } \Vert u\Vert _{\infty }. \end{aligned}$$

Next we want to get rid of the $\Vert u\Vert _{\infty }$ term. For that we note that there exists $s_0\in [0, 2\pi ]$ satisfying $|u^2(s_0)| \le \frac{1}{2\pi }\Vert u\Vert _2^2$. We use this in the following way,

$$\begin{aligned} \Vert u\Vert _{\infty }^2&\le |u^2(s_0)|+\sup _{s\in [0,2\pi ]} |u^2(s)-u^2(s_0)| \le \frac{1}{2\pi } \Vert u\Vert _2^2+\int _0^{2\pi } \, 2|u| |u'| \, ds \\&\le \frac{1}{2\pi }\Vert u\Vert _2^2+2\Vert u\Vert _2\Vert u'\Vert _2 {\mathop {\le }\limits ^{(14)}} \frac{F}{2\pi }\Vert u\Vert _2+2\Vert u\Vert _2\Vert u'\Vert _2 \\&\le \Vert u\Vert _2\bigg (\frac{F}{2\pi } +1+\Vert u'\Vert _2^2\bigg ) , \end{aligned}$$

from where we find

$$\begin{aligned} \Vert u\Vert _{\infty } \le \Vert u\Vert _2^{\frac{1}{2}} \bigg (\sqrt{\frac{F}{2\pi }} +1+\Vert u'\Vert _2\bigg ). \end{aligned}$$

In total, we have

$$\begin{aligned} \Vert u'\Vert _2^2\le & {} \frac{\Vert f\Vert _2^2}{2|d|} \Vert u\Vert _2 \Vert u'\Vert _2 +\Vert f''\Vert _2 \Vert u\Vert _2 +2\Vert f'\Vert _{\infty } \Vert u\Vert _2^{\frac{1}{2}} \bigg (\sqrt{\frac{F}{2\pi }} +1+\Vert u'\Vert _2\bigg ) \\&{\mathop {\le }\limits ^{(14)}}&\frac{F^{\frac{11}{4}}}{2|d|} \Vert u\Vert _2^{\frac{1}{4}} \Vert u'\Vert _2 +\Vert f''\Vert _2 F^{\frac{1}{2}} \Vert u\Vert _2^{\frac{1}{2}} +2\Vert f'\Vert _{\infty } \Vert u\Vert _2^{\frac{1}{2}} \bigg (\sqrt{\frac{F}{2\pi }} +1\bigg ) +2\Vert f'\Vert _{\infty } F^{\frac{1}{4}} \Vert u\Vert _2^{\frac{1}{4}} \Vert u'\Vert _2 \\= & {} \bigg (\frac{F^{\frac{11}{4}}}{2|d|}+2\Vert f'\Vert _{\infty } F^{\frac{1}{4}}\bigg ) \Vert u\Vert _2^{\frac{1}{4}} \Vert u'\Vert _2 +\bigg (\Vert f''\Vert _2 F^{\frac{1}{2}}+2\Vert f'\Vert _{\infty }\bigg (\sqrt{\frac{F}{2\pi }} +1\bigg )\bigg )\Vert u\Vert _2^{\frac{1}{2}} \\=: & {} A_1\Vert u\Vert _2^{\frac{1}{4}} \Vert u'\Vert _2 +A_2^2 \Vert u\Vert _2^{\frac{1}{2}}. \end{aligned}$$

This is a quadratic inequality in $\Vert u'\Vert _2$ which implies

$$\begin{aligned} \Vert u'\Vert _2 \le \frac{A_1\Vert u\Vert _2^{\frac{1}{4}}+\sqrt{A_1^2\Vert u\Vert _2^{\frac{1}{2}}+4 A_2^2\Vert u\Vert _2^{\frac{1}{2}}}}{2} \le A_1\Vert u\Vert _2^{\frac{1}{4}}+A_2 \Vert u\Vert _2^{\frac{1}{4}}=B \Vert u\Vert _2^{\frac{1}{4}} \end{aligned}$$

as claimed.

Step 3. Here, we prove

$$\begin{aligned} \Vert u\Vert _{\infty }\le C. \end{aligned}$$

(18)

There exists $s_1 \in [0,2\pi ]$ satisfying $|u(s_1)|\le \frac{\Vert u\Vert _2}{\sqrt{2\pi }}$. The claim now follows from

$$\begin{aligned} \Vert u\Vert _{\infty } \le&|u(s_1)|+\sup _{s\in [0,2\pi ]} |u(s)-u(s_1)| \le \frac{\Vert u\Vert _2}{\sqrt{2\pi }}+\Vert u'\Vert _1 \le \frac{\Vert u\Vert _2}{\sqrt{2\pi }}+\sqrt{2\pi }\Vert u'\Vert _2 \\ {\mathop {\le }\limits ^{(14), (17)}}&\left( \frac{F^\frac{3}{4}}{\sqrt{2\pi }}+\sqrt{2\pi }B\right) \Vert u\Vert _2^\frac{1}{4} {\mathop {\le }\limits ^{(14)}} C. \end{aligned}$$

Step 4. Next we show in the case $\zeta {\text {sign}}(d)<-C^2 {\textbf{1}}_{d<0}$ the additional $L^2$-bound

$$\begin{aligned} \Vert u\Vert _2 \le D. \end{aligned}$$

(19)

After integrating (15) over $[0,2\pi ]$ and taking the real part of the resulting equation, we get

$$\begin{aligned} d \Vert u'\Vert _2^2 =\omega \int _0^{2\pi } \, {\text {Im}}(u'{\bar{u}}) \, ds -\zeta \Vert u\Vert _2^2 +\Vert u\Vert _4^4 +{\text {Im}}\int \limits _0^{2\pi } \, f {\bar{u}}\, ds. \end{aligned}$$

In order to prove (19), we first suppose $d>0$. Then, we have on one hand

$$\begin{aligned} d \Vert u'\Vert _2^2 {\mathop {\le }\limits ^{(17)}} d B^2 \Vert u\Vert _2^{\frac{1}{2}} \end{aligned}$$

(20)

and on the other hand

$$\begin{aligned} \begin{aligned} \omega \int _0^{2\pi } \, {\text {Im}}(u'{\bar{u}}) \, ds -\zeta \Vert u\Vert _2^2 +\Vert u\Vert _4^4 +{\text {Im}}\int \limits _0^{2\pi } \, f{\bar{u}}\, ds&\ge -|\omega | \Vert u\Vert _2\Vert u'\Vert _2 -\zeta \Vert u\Vert _2^2 -F\Vert u\Vert _2 \\&{\mathop {\ge }\limits ^{(17)}} -|\omega |B\Vert u\Vert _2^{\frac{5}{4}} -\zeta \Vert u\Vert _2^2 -F\Vert u\Vert _2 \\&{\mathop {\ge }\limits ^{(14)}} -|\omega |B F^{\frac{3}{4}}\Vert u\Vert _2^{\frac{1}{2}} -\zeta \Vert u\Vert _2^2 -F^{\frac{3}{2}}\Vert u\Vert _2^{\frac{1}{2}}. \end{aligned} \end{aligned}$$

(21)

Combining the two estimates (20), (21) and grouping quadratic terms and terms of power $\frac{1}{2}$ of $\Vert u\Vert _2$ on separate sides of the inequality, we get

$$\begin{aligned} -\zeta \Vert u\Vert _2^2 \le \Bigl (F^{\frac{3}{2}}+ |\omega |B F^{\frac{3}{4}} +dB^2\Bigr )\Vert u\Vert _2^{\frac{1}{2}} \end{aligned}$$

which finally implies $\Vert u\Vert _2 \le D$ whenever $\zeta <0$. Assuming now $d<0$ the estimate (20) becomes

$$\begin{aligned} d \Vert u'\Vert _2^2 \ge -|d| B^2 \Vert u\Vert _2^{\frac{1}{2}} \end{aligned}$$

(22)

whereas in (21) the term $\Vert u\Vert _4^4$, which was previously dropped, now has to be estimated by $\Vert u\Vert _4^4\le \Vert u\Vert _\infty ^2\Vert u\Vert _2^2\le C^2\Vert u\Vert _2^2$. The estimate (21) now becomes

$$\begin{aligned} \omega \int _0^{2\pi } \, {\text {Im}}(u'{\bar{u}}) \, ds -\zeta \Vert u\Vert _2^2 +\Vert u\Vert _4^4 +{\text {Im}}\int \limits _0^{2\pi } \, f {\bar{u}} \, ds \le |\omega |B F^{\frac{3}{4}}\Vert u\Vert _2^{\frac{1}{2}} +(C^2-\zeta ) \Vert u\Vert _2^2 +F^{\frac{3}{2}}\Vert u\Vert _2^{\frac{1}{2}}. \end{aligned}$$

(23)

The combination of (22) and (23) leads to

$$\begin{aligned} (\zeta -C^2)\Vert u\Vert _2^2 \le \Bigl (F^{\frac{3}{2}}+ |\omega |B F^{\frac{3}{4}} +|d|B^2\Bigr )\Vert u\Vert _2^{\frac{1}{2}} \end{aligned}$$

which again implies $\Vert u\Vert _2 \le D$ whenever $-\zeta <-C^2$.

Step 5. Finally, we prove

$$\begin{aligned} \Vert u\Vert _{\infty }\le \biggl (\frac{F^{\frac{3}{4}}}{\sqrt{2\pi }}+\sqrt{2\pi }B\biggr )D^{\frac{1}{4}} \end{aligned}$$

(24)

whenever $\zeta {\text {sign}}(d)<-C^2 {\textbf{1}}_{d<0}$. For this we repeat Step 3 and use in the final estimate that $\Vert u\Vert _2\le D$. $\square$

5 Proof of existence (Theorem 1) and uniqueness (Theorem 7) statements

Let us consider the operator $L:H^2_{\text {per}}(0,2\pi ) \rightarrow L^2(0,2\pi )$ with $Lu=L_0u -\textrm{i}u$ and $L_0 u=-d u''+\textrm{i}\omega u'+\zeta u$. Since $L_0:H^2_{\text {per}}(0,2\pi ) \rightarrow L^2(0,2\pi )$ is self-adjoint its spectrum is real and we see that L has spectrum on the line $-\textrm{i}+\mathbb {R}$. In particular, L is invertible and $L^{-1}:L^2(0,2\pi )\rightarrow H^2_{\text {per}}(0,2\pi )$ is bounded. By using the compact embedding $H^2_{\text {per}}(0,2\pi ) \hookrightarrow H^1_{\text {per}}(0,2\pi )$ we see that

$$\begin{aligned} L^{-1}:L^2(0,2\pi ) \rightarrow H^1_{\text {per}}(0,2\pi ) \text{ is } \text{ compact }. \end{aligned}$$

Since moreover $H^1_{\text {per}}(0,2\pi )$ is a Banach algebra we can rewrite (3) as a fixed point problem $u=\Phi (u)$, where $\Phi$ denotes the compact map

$$\begin{aligned} \Phi :H^1_{\text {per}}(0,2\pi ) \rightarrow H^1_{\text {per}}(0,2\pi ), \ \Phi (u)=L^{-1}\big (|u|^2 u-\textrm{i}f(s)\big ). \end{aligned}$$

In order to prove our first existence result from Theorem 1, let us recall Schaefer’s fixed point theorem ([5, Corollary 8.1]).

Theorem 6

(Schaefer’s fixed point theorem) Let X be a Banach space and $\Phi :X \rightarrow X$ be compact. Suppose that the set

$$\begin{aligned} \{x \in X: \, x=\lambda \Phi (x) \text { for some } \lambda \in (0,1)\} \end{aligned}$$

is bounded. Then, $\Phi$ has a fixed point.

Proof of Theorem 1

Let $u\in H^1_{\text {per}}(0,2\pi )$ and $u=\lambda \Phi (u)$ for some $\lambda \in (0,1)$. Then, $u\in H^2_{\text {per}}(0,2\pi )$ and

$$\begin{aligned} -d u''+\textrm{i} \omega u'+(\zeta -\textrm{i})u-\lambda |u|^2 u+\textrm{i}\lambda f(s)=0. \end{aligned}$$

Let us now define $v \in H^2_{\text {per}}(0,2\pi )$ by $v(s)=\sqrt{\lambda }u(s)$. Then

$$\begin{aligned}-d v''+\textrm{i} \omega v'+(\zeta -\textrm{i})v-|v|^2 v+\textrm{i} {\tilde{f}}(s)=0 \end{aligned}$$

with ${\tilde{f}}=\lambda ^\frac{3}{2} f$. Estimate (11) of Theorem 5 with ${\tilde{F}}=F(\lambda ^\frac{3}{2} f)=\lambda ^\frac{3}{2}F$ implies

$$\begin{aligned} \Vert u\Vert _2 = \frac{1}{\sqrt{\lambda }} \Vert v\Vert _2 \le \frac{1}{\sqrt{\lambda }} {\tilde{F}}=\lambda F\le F. \end{aligned}$$

Using (12) from Theorem 5 with $\tilde{B}=B(d,\lambda ^\frac{3}{2} f)$, we also find

$$\begin{aligned} \Vert u'\Vert _2&= \frac{1}{\sqrt{\lambda }} \Vert v'\Vert _2 \le \frac{1}{\sqrt{\lambda }}{\tilde{B}} {\tilde{F}}^{\frac{1}{4}} \\&= \lambda ^4\frac{F^3}{2|d|}+2\lambda ^{\frac{7}{4}}\Vert f'\Vert _{\infty } F^{\frac{1}{2}}+\sqrt{\lambda ^2 \Vert f''\Vert _2 F+2\lambda ^{\frac{5}{4}}\Vert f'\Vert _{\infty } \biggl (\frac{\lambda ^{\frac{3}{4}} F}{\sqrt{2\pi }}+\sqrt{F}\biggr )} \\&\le \frac{F^3}{2|d|}+2\Vert f'\Vert _{\infty } F^{\frac{1}{2}}+\sqrt{\Vert f''\Vert _2 F+2\Vert f'\Vert _{\infty } \biggl (\frac{F}{\sqrt{2\pi }}+\sqrt{F}\biggr )}=B F^{\frac{1}{4}}. \end{aligned}$$

The assertion now follows from Theorem 6. $\square$

For the next uniqueness result, cf. Theorem 7, let us rewrite the constant D from Theorem 5 as

$$\begin{aligned} D=D(d,f,\omega ,\zeta )=\biggl (\frac{{\tilde{D}}}{(-\zeta {\text {sign}}(d)-C^2 {\textbf{1}}_{d<0})_+}\biggr )^{\frac{2}{3}} \end{aligned}$$

with

$$\begin{aligned} {\tilde{D}}={\tilde{D}}(d,f,\omega ) = F^{\frac{3}{2}}+ |\omega |B F^{\frac{3}{4}} +|d|B^2. \end{aligned}$$

Our result complements the existence statement provided in Theorem 1 by a uniqueness statement. It consists of three cases: (i) and (ii) cover the case where $|\zeta |\gg 1$ is sufficiently large, whereas (iii) builds upon $\Vert f\Vert \ll 1$ measured in a suitable norm $\Vert \cdot \Vert$ such that the constant $C=C(d,f)$ becomes small. This is the case, e.g., if $\Vert f\Vert _2\ll 1$ and $\Vert f''\Vert _2$ remains bounded.

Theorem 7

Let $d \in \mathbb {R}{\setminus }\{0\}$, $\zeta ,\omega \in \mathbb {R}$ and $f \in H^2(0,2\pi )$. Then, (3) has a unique solution $u \in H^2_{\text {per}}(0,2\pi )$ in the following three cases,

• (i)
  $$\begin{aligned} {{\,\textrm{sign}\,}}(d)\zeta <\zeta _{*}, \end{aligned}$$
• (ii)
  $$\begin{aligned} {{\,\textrm{sign}\,}}(d)\zeta >\zeta ^*, \end{aligned}$$
• (iii)
  $$\begin{aligned} \sqrt{3}C<1, \end{aligned}$$

where $\zeta _{*}\le 0 \le \zeta ^*$ are given by

$$\begin{aligned} \zeta _{*}&=\zeta _{*}(d,f,\omega )=-C^2 {\textbf{1}}_{d<0} -\frac{27(F^{\frac{3}{4}}+2\pi B)^6 {\tilde{D}}}{8\pi ^3}, \\ \zeta ^*&=\zeta ^*(d,f,\omega )=3C^2+\frac{\omega ^2}{4|d|}, \end{aligned}$$

and $F=F(f)$, $B=B(d,f)$, $C=C(d,f)$ are the constants from Theorem 5.

Proof

It suffices to consider the case $f \ne 0$. By Theorem 1 we know that (3) has at least one solution $u_1\in H^2_{\text {per}}(0,2\pi )$. Now let $u_2\in H^2_{\text {per}}(0,2\pi )$ denote an additional solution and define

$$\begin{aligned} R=R(d,f,\omega ,\zeta )={\left\{ \begin{array}{ll} \min \biggl \{C,\biggl (\frac{F^{\frac{3}{4}}}{\sqrt{2\pi }}+\sqrt{2\pi }B\biggr )D^{\frac{1}{4}}\biggr \}, &{} \zeta {\text {sign}}(d)+C^2 {\textbf{1}}_{d<0}<0, \\ C, &{} \zeta {\text {sign}}(d)+C^2 {\textbf{1}}_{d<0}\ge 0. \end{array}\right. } \end{aligned}$$

Then $\Vert u_j\Vert _{\infty }\le R$ for $j=1,2$ by Theorem 5, which easily implies

$$\begin{aligned} \big \Vert |u_1|^2 u_1-|u_2|^2u_2\big \Vert _2 \le 3R^2 \Vert u_1-u_2\Vert _2. \end{aligned}$$

Since $u_j, j=1,2$ solves the fixed point problem $u_j=\Phi (u_j)$ we obtain

$$\begin{aligned} \Vert u_1-u_2\Vert _2=\Vert \Phi (u_1)-\Phi (u_2)\Vert _2 \le 3R^2\Vert L^{-1}\Vert \Vert u_1-u_2\Vert _2, \end{aligned}$$

where $\Vert L^{-1}\Vert =\sup _{v \in L^2(0,2\pi ), \Vert v\Vert _2=1} \Vert L^{-1}v\Vert _2$. Next we show $3R^2\Vert L^{-1}\Vert <1$ which implies $u_1=u_2$ and thus finishes the proof. To this end we decompose a function $v\in L^2(0,2\pi )$ into its Fourier series, i.e., $v=\sum _{m\in \mathbb {Z}} v_m \textrm{e}^{\textrm{i}ms}$ so that

$$\begin{aligned} L^{-1}v=\sum \limits _{m\in \mathbb {Z}} \frac{v_m}{dm^2-\omega m+\zeta -\textrm{i}}\, \textrm{e}^{\textrm{i}ms}. \end{aligned}$$

On one hand we get $\Vert L^{-1}\Vert \le 1$ since

$$\begin{aligned} \Vert L^{-1}v\Vert _2^2=2\pi \sum \limits _{m\in \mathbb {Z}} \frac{|v_m|^2}{1+(d m^2-\omega m+\zeta )^2} \le 2\pi \sum \limits _{m\in \mathbb {Z}}|v_m|^2=\Vert v\Vert _2^2. \end{aligned}$$

On the other hand, if ${\text {sign}}(d)\big (\zeta -\frac{\omega ^2}{4d}\big )>0$, we get

$$\begin{aligned} \Vert L^{-1}v\Vert _2^2&=2\pi \sum \limits _{m\in \mathbb {Z}} \frac{|v_m|^2}{1+(d m^2-\omega m+\zeta )^2} =2\pi \sum \limits _{m\in \mathbb {Z}} \frac{|v_m|^2}{1+\Big (d\big (m-\frac{\omega }{2d}\big )^2+\zeta -\frac{\omega ^2}{4d}\Big )^2} \\&\le 2\pi \sum \limits _{m\in \mathbb {Z}} \frac{|v_m|^2}{\big (\zeta -\frac{\omega ^2}{4d}\big )^2}=\frac{1}{\big (\zeta -\frac{\omega ^2}{4d}\big )^2} \Vert v\Vert _2^2, \end{aligned}$$

i.e., $\Vert L^{-1}\Vert \le {\text {sign}}(d)\big (\zeta -\frac{\omega ^2}{4d}\big )^{-1}$.

In case (i) where ${\text {sign}}(d)\zeta<\zeta _*<-C^2 {\textbf{1}}_{d<0} \le {0}$, we use $\Vert L^{-1}\Vert \le 1$ and find by the definition of R and $\zeta _*$ that

$$\begin{aligned} 3R^2 \Vert L^{-1}\Vert&\le 3 \frac{(F^{\frac{3}{4}}+2\pi B)^2}{2\pi } D^{\frac{1}{2}} \\&= 3 \frac{(F^{\frac{3}{4}}+2\pi B)^2}{2\pi } \biggl (\frac{{\tilde{D}}}{-\zeta {\text {sign}}(d)-C^2 {\textbf{1}}_{d<0}}\biggr )^{\frac{1}{3}} \\&< 3 \frac{(F^{\frac{3}{4}}+2\pi B)^2}{2\pi } \biggl (\frac{\tilde{D}}{-\zeta _* -C^2 {\textbf{1}}_{d<0}}\biggr )^{\frac{1}{3}}=1. \end{aligned}$$

In case (ii) where ${\text {sign}}(d)\zeta>\zeta ^* > \frac{\omega ^2}{4|d|}\ge 0$, we use $\Vert L^{-1}\Vert \le {\text {sign}}(d)\big (\zeta -\frac{\omega ^2}{4d}\big )^{-1}$ and get by the choice of $\zeta ^*$

$$\begin{aligned} 3R^2 \Vert L^{-1}\Vert \le \frac{3 C^2}{{\text {sign}}(d)(\zeta -\frac{\omega ^2}{4d})} <\frac{3C^2}{\zeta ^*-\frac{\omega ^2}{4|d|}}=1. \end{aligned}$$

In case (iii) where $\sqrt{3}C<1$, we use $\Vert L^{-1}\Vert \le 1$ to conclude

$$\begin{aligned} 3R^2 \Vert L^{-1}\Vert \le 3 C^2 <1. \end{aligned}$$

$\square$

6 Proof of the continuation results

In this section, we continue to use the notion for the operator $L:H^2_{\text {per}}(0,2\pi ) \rightarrow L^2(0,2\pi )$ from Sect. 4. We also use that $L^{-1}:L^2(0,2\pi ) \rightarrow H^2_{\text {per}}(0,2\pi )$ is bounded and that $L^{-1}:L^2(0,2\pi ) \rightarrow H^1_{\text {per}}(0,2\pi )$ is compact. We first consider continuation from a trivial solution. In order to prove Theorem 2, let us provide the following global continuation theorem.

Theorem 8

Let X be a real Banach space and $K \in C^1(\mathbb {R}\times X,X)$ be compact. We consider the problem

$$\begin{aligned} T(\lambda ,x){:}{=}x-K(\lambda ,x)=0. \end{aligned}$$

(25)

Assume that $T(\lambda _0,x_0)=0$ and that $\partial _x T(\lambda _0,x_0)$ is invertible. Then, there exists a connected and closed set (=continuum) ${\mathcal {C}}^+ \subset [\lambda _0,\infty ) \times X$ of solutions of (25) with $(\lambda _0,x_0)\in {\mathcal {C}}^+$. For ${\mathcal {C}}^+$, one of the following alternatives holds:

• (a) ${\mathcal {C}}^+$ is unbounded, or

• (b) $\exists x_0^+ \in X\setminus \{x_0\}: \, (\lambda _0,x_0^+) \in {\mathcal {C}}^+.$

If one chooses ${\mathcal {C}}^+$ to be maximally connected then there is no more a strict alternative between (a) and (b) and instead at least one of the two (possibly both) properties holds.

Remark 7

($\alpha$) The theorem follows from [2, Theorem 3.3] or [26, Theorem 1.3.2] because the invertibility of $\partial _x T(\lambda _0,x_0)$ implies that ${\text {deg}}(T(\lambda _0,\cdot ),B_{\varepsilon }(x_0),0)={\text {deg}}(\partial _x T(\lambda _0,x_0),B_{\varepsilon }(0),0) \ne 0$.

($\beta$) There exists also a continuum ${\mathcal {C}}^- \subset (-\infty ,\lambda _0] \times X$ of solutions of (25) with $(\lambda _0,x_0)\in {\mathcal {C}}^-$ satisfying one of the alternatives of the theorem.

($\gamma$) Alternative (a) of Theorem 8 means that ${\mathcal {C}}^+$ is unbounded either in the Banach space direction X or in the parameter direction $[\lambda _0,\infty )$ or in both. If unboundedness in the Banach space direction is excluded on compact intervals $[\lambda _0,\Lambda ]$, e.g., by a-priori bounds, then unboundedness in the parameter direction follows, i.e., the projection of ${\mathcal {C}}^+$ onto $[\lambda _0,\infty )$ denoted by ${\text {pr}}_1({\mathcal {C}}^+)$ must coincide with $[\lambda _0,\infty )$. This is an existence result for all $\lambda \ge \lambda _0$ which is one aspect of Theorem 2.

($\delta$) Alternative (b) of Theorem 8 means that the continuum ${\mathcal {C}}^+$ returns to the $\lambda =\lambda _0$ line at a point $x_0^+\not = x_0$.

Proof of Theorem 2

We define the map $K:\mathbb {R}\times H^1_{\text {per}}(0,2\pi ) \rightarrow H^1_{\text {per}}(0,2\pi )$ as $K(f_1,u){:}{=}L^{-1}(|u|^2 u-\textrm{i}f_0-\textrm{i}f_1 e(s))$ and set $T(f_1,u){:}{=}u-K(f_1,u)$. Then, as explained before Theorem 6, K is compact and

$$\begin{aligned} T(0,u_0)=u_0-L^{-1}(|u_0|^2 u_0-\textrm{i}f_0){\mathop {=}\limits ^{(4)}}u_0-L^{-1}\big ((\zeta -\textrm{i})u_0\big )=u_0-u_0=0. \end{aligned}$$

Next we show that $\partial _u T(0,u_0)$ is invertible. To this end note that

$$\begin{aligned} \begin{aligned} \partial _u T(0,u_0) \varphi =\varphi -L^{-1} (2|u_0|^2 \varphi +u_0^2 \overline{\varphi }) \quad \text{ for } \varphi \in H^1_{\text{ per }}(0,2\pi ) \end{aligned} \end{aligned}$$

and hence, as a compact perturbation of the identity, $\partial _u T(0,u_0)$ is invertible if it is injective. Since $u_0$ is constant this amounts exactly to the characterization of non-degeneracy of $u_0$ as described in Lemma 1.

Now assertion (i) follows from the classical implicit function theorem and Theorem 8 yields that the maximal continuum ${\mathcal {C}}^+ \subset [0,\infty ) \times H^1_{\text {per}}(0,2\pi )$ of solutions $(f_1,u)$ of (3) with $(0,u_0)\in {\mathcal {C}}^+$ is unbounded or returns to another solution at $f_1=0$. The continuum ${\mathcal {C}}^+$ in fact belongs to $[0,\infty )\times H^2_{\text {per}}(0,2\pi )$ and persists as a connected and closed set in the stronger topology of $[0,\infty )\times H^2_{\text {per}}(0,2\pi )$. Next we show that the unboundedness of ${\mathcal {C}}^+$ coincides with ${\text {pr}}_1({\mathcal {C}}^+)=[0,\infty )$. According to Remark 7.($\gamma$), we need to show that unboundedness in the Banach space direction $H^1_\text {per}(0,2\pi )$ is excluded for $f_1$ in bounded intervals. To see this suppose that $0\le f_1 \le M$ for all $(f_1,u)\in {\mathcal {C}}^+$ and some constant $M>0$. Then, by the a-priori bounds (11) and (12) from Theorem 5 we get

$$\begin{aligned} \Vert u\Vert _2 \le \Vert f_0+f_1 e(s) \Vert _2 \le \sqrt{2\pi }|f_0|+M\Vert e\Vert _2=:N=N(f_0,M,e) \end{aligned}$$

and

$$\begin{aligned} \Vert u'\Vert _2 \le \frac{N^3}{2|d|}+2M \Vert e'\Vert _{\infty } N^{\frac{1}{2}} +\sqrt{M\Vert e''\Vert _2 N+2M\Vert e'\Vert _{\infty }\biggl (\frac{N}{\sqrt{2\pi }}+\sqrt{N}\biggr )} \end{aligned}$$

for all $(f_1,u)\in {\mathcal {C}}^+$. Hence, ${\mathcal {C}}^+$ is bounded in the Banach space direction. Assertion (ii) follows in a similar way by using the a-priori bounds of Theorem 5 and the fact that by (3) the bounds for $\Vert u\Vert _2$, $\Vert u'\Vert _2$ and $\Vert u\Vert _{\infty }$ translate into a bound for $\Vert u''\Vert _2$.

According to Remark 7.($\beta$), the above line of arguments also yields that the maximal continuum ${\mathcal {C}}^- \subset (-\infty ,0] \times H^2_{\text {per}}(0,2\pi )$ of solutions of (3) with $(0,u_0)\in {\mathcal {C}}^-$ satisfies ${\text {pr}}_1({\mathcal {C}}^-)=(-\infty ,0]$ or returns to another solution at $f_1=0$. This finishes the proof. $\square$

Proof of Corollary 1

The result follows from a combination of Theorems 2 and  7. For $f_1=0$, i.e., $f(s)= f_0$, the abbreviations F, B, C from Theorem 5 and ${\tilde{D}}$ from Theorem 7 reduce to

$$\begin{aligned} F(f_0)&=\sqrt{2\pi }|f_0|, \quad B(d,f_0)=2^{\frac{3}{8}} \pi ^{\frac{11}{8}} |f_0|^{\frac{11}{4}} |d|^{-1}, \\ C(d,f_0)&=|f_0|(1+2\pi ^2 f_0^2 |d|^{-1}), \\ {\tilde{D}}(d,f_0,\omega )&=(2\pi )^{\frac{3}{4}}|f_0|^{\frac{3}{2}} (|d|+\pi f_0^2 |\omega | +\pi ^2 f_0^4)|d|^{-1}. \\ \end{aligned}$$

Hence, the constants $\zeta _*, \zeta ^*$ from Theorem 7 take the form

$$\begin{aligned} \zeta _{*}(d,f_0,\omega )&=-C^2(d,f_0) {\textbf{1}}_{d<0}- 27\biggl (1+\frac{\pi f_0^2 |\omega |}{|d|}+\frac{\pi ^2 f_0^4}{|d|}\biggr ) C(d,f_0)^6, \\ \zeta ^*(d,f_0,\omega )&= 3C(d,f_0)^2+\frac{\omega ^2}{4|d|}. \end{aligned}$$

Finally, the conditions (i), (ii), (iii) from the uniqueness result of Theorem 7 translate into the conditions (i), (ii), (iii) from Corollary 1. $\square$

Now we turn to continuation from a non-trivial solution. Theorem 3 will follow from the theorem of Crandall–Rabinowitz on bifurcation from a simple eigenvalue, which we recall next.

Theorem 9

(Crandall–Rabinowitz [4, 16]) Let $I\subset \mathbb {R}$ be an open interval, X,Y Banach spaces and let $F:I\times X\rightarrow Y$ be twice continuously differentiable such that $F(\lambda ,0)=0$ for all $\lambda \in I$ and $\partial _x F(\lambda _0,0):X\rightarrow Y$ is an index-zero Fredholm operator for $\lambda _0\in I$. Moreover assume:

• (H1) there is $\phi \in X,\phi \ne 0$ such that $\ker \partial _x F(\lambda _0,0)={{\,\textrm{span}\,}}\{\phi \}$,

• (H2) $\partial ^2_{x,\lambda } F(\lambda _0,0)[\phi ] \not \in {{\,\textrm{range}\,}}\partial _xF(\lambda _0,0)$.

Then, there exists $\epsilon >0$ and a continuously differentiable curve $(\lambda ,x): (-\epsilon ,\epsilon )\rightarrow I\times X$ with $\lambda (0)=\lambda _0$, $x(0)=0$, $\dot{x}(0)=\phi$ and $x(t)\ne 0$ for $0<|t|<\epsilon$ and $F(\lambda (t),x(t))=0$ for all $t\in (-\epsilon ,\epsilon )$. Moreover, there exists a neighborhood $J\times U\subset I\times X$ of $(\lambda _0,0)$ such that all non-trivial solutions in $J\times U$ of $F(\lambda ,x)=0$ lie on the curve. Finally,

$$\begin{aligned} \dot{\lambda }(0)= -\frac{1}{2} \frac{\langle \partial ^2_{xx} F(\lambda _0,0)[\phi ,\phi ],\phi ^*\rangle }{\langle \partial ^2_{x,\lambda } F(\lambda _0,0)[\phi ],\phi ^*\rangle }, \end{aligned}$$

where ${{\,\textrm{span}\,}}\{\phi ^*\}=\ker \partial _x F(\lambda _0,0)^*$ and $\langle \cdot ,\cdot \rangle$ is the duality pairing between Y and its dual $Y^*$.

Next we provide the functional analytic setup. Fix the values of $d, \omega , \zeta , f_0$ and the function e. If $u_0\in H^2_\text {per}(0,2\pi )$ is the non-trivial non-degenerate solution of (3) for $f_1=0$ (as assumed in Theorem 3) then for $\sigma \in \mathbb {R}$ we denote by $u_\sigma (s){:}{=}u_0(s-\sigma )$ its shifted copy, which is also a solution of (3) for $f_1=0$. Consider the mapping

$$\begin{aligned} G:\left\{ \begin{array}{rcl} \mathbb {R}\times H^2_{\text {per}(0,2\pi )} &{} \rightarrow &{} L^2(0,2\pi ), \\ (f_1,u) &{} \mapsto &{} -d u''+\textrm{i} \omega u'+(\zeta -\textrm{i})u-|u|^2 u+\textrm{i}f_0+\textrm{i}f_1 e(s). \end{array} \right. \end{aligned}$$

Then, G is twice continuously differentiable. The linearized operator $\partial _{(f_1,u)} G(0,u_\sigma )=(\textrm{i}e,L_{u_\sigma })$ with $L_{u_\sigma }$ as in Definition 1 is a Fredholm operator with the property that $(0,u_\sigma ')\in \ker \partial _{(f_1,u)} G(0,u_\sigma )$. As we shall see there may be more elements in the kernel. Next we fix the value $\sigma _0$ (its precise value will be given later) and decompose $H^2_\text {per}(0,2\pi ) = {{\,\textrm{span}\,}}\{u_{\sigma _0}'\}\oplus Z$ where, e.g.,

$$\begin{aligned} Z {:}{=}H^2_\text {per}(0,2\pi ) \cap {{\,\textrm{span}\,}}\{u_{\sigma _0}'\}^{\perp _{L^2}} =\biggl \{\varphi -\frac{\langle \varphi , u_{\sigma _0}'\rangle _{L^2}}{\langle u_{\sigma _0}', u_{\sigma _0}'\rangle _{L^2}} u_{\sigma _0}': \varphi \in H^2_\text {per}(0,2\pi )\biggr \}. \end{aligned}$$

It will be more convenient to rewrite $u= u_\sigma +v$ with $v\in Z$. In order to justify this, note also that the map $(\sigma ,v)\mapsto u_{\sigma }+v$ defines a diffeomorphism of a neighborhood of $(\sigma _0,0) \in \mathbb {R}\times Z$ onto a neighborhood of $u_{\sigma _0} \in H^2_{\text {per}}(0,2\pi )$ since the derivative at $(\sigma _0,0)$ is given by $(\lambda ,\psi ) \mapsto -\lambda u_{\sigma _0}'+\psi$ which is an isomorphism from $\mathbb {R}\times Z$ onto $H^2_{\text {per}}(0,2\pi )$. Now we define

$$\begin{aligned} F:\left\{ \begin{array}{rcl} \mathbb {R}\times \mathbb {R}\times Z \ {} &{} \rightarrow &{} L^2(0,2\pi ), \\ (\sigma ,f_1,v) &{} \mapsto &{} G(f_1,u_\sigma +v) \end{array} \right. \end{aligned}$$

which is also twice continuously differentiable and where $\partial _{(f_1,v)} F(\sigma _0,0,0)$ is a Fredholm operator with index zero. Our goal will be to solve

$$\begin{aligned} F(\sigma ,f_1,v)=0 \end{aligned}$$

(26)

by means of bifurcation theory, where $\sigma \in \mathbb {R}$ is the bifurcation parameter. Notice that $F(\sigma ,0,0)=0$ for all $\sigma \in \mathbb {R}$, i.e., $(f_1,v)=(0,0)$ is a trivial solution of (26).

Next we show (H1) of Theorem 9.

Lemma 2

Suppose that $\sigma _0 \in \mathbb {R}$ satisfies (7), i.e., ${\text {Im}}\int _0^{2\pi }e(s+\sigma _0)\overline{\phi _0^*(s)}\,ds =0$. Then, we have that $\dim \ker \partial _{(f_1,v)} F(\sigma _0,0,0)=1$ and ${{\,\textrm{range}\,}}\partial _{(f_1,v)} F(\sigma _0,0,0) = {{\,\textrm{span}\,}}\{\phi _{\sigma _0}^*\}^{\perp _{L^2}}$.

Proof

The fact that $\partial _{(f_1,v)} F(\sigma _0,0,0)$ is a Fredholm operator follows from Remark 1. For $(\alpha ,\psi _{\sigma _0})\in \mathbb {R}\times Z$ being non-trivial and belonging to the kernel of $\partial _{(f_1,v)} F(\sigma _0,0,0)$, we have

$$\begin{aligned} \partial _{(f_1,v)} F(\sigma _0,0,0)[\alpha ,\psi _{\sigma _0}]= L_{u_{\sigma _0}}\psi _{\sigma _0}+\textrm{i}\alpha e=0. \end{aligned}$$

(27)

If $\alpha =0$ then by non-degeneracy we find $\psi _{\sigma _0}\in {{\,\textrm{span}\,}}\{u_{\sigma _0}'\}\cap Z=\{0\}$, which is impossible. Hence, we may assume w.l.o.g. that $\alpha =1$ and $\psi _{\sigma _0}$ has to solve

$$\begin{aligned} L_{u_{\sigma _0}}\psi _{\sigma _0} = -\textrm{i}e \end{aligned}$$

(28)

which, by setting $\psi _{\sigma _0}(s)=\xi _{\sigma _0}(s-\sigma _0)$, is equivalent to

$$\begin{aligned} L_{u_0}\xi _{\sigma _0} = -\textrm{i}e(\cdot +\sigma _0). \end{aligned}$$

(29)

By the Fredholm alternative, this is possible if and only if $-\textrm{i}e(\cdot +\sigma _0)\perp _{L^2} \phi _0^*$. If this $L^2$-orthogonality holds then there exists $\psi _{\sigma _0}\in H^2_\text {per}(0,2\pi )$ solving (28) and $\psi _{\sigma _0}$ is unique up to adding a multiple of $u_{\sigma _0}'$. Hence, there is a unique $\psi _{\sigma _0}\in Z$ solving (28). The $L^2$-orthogonality means

$$\begin{aligned} 0=-{\text {Re}}\int _0^{2\pi }\textrm{i}e(s+\sigma _0) \overline{\phi _0^*(s)}\,ds = {\text {Im}}\int \limits _0^{2\pi } e(s+\sigma _0) \overline{\phi _0^*(s)}\,ds \end{aligned}$$

which amounts to (7). Finally, it remains to determine the range of $\partial _{(f_1,v)} F(\sigma _0,0,0)$. Let $\tilde{\phi }\in L^2(0,2\pi )$ be such that $\tilde{\phi }= \partial _{(f_1,v)} F(\sigma _0,0,0)[\alpha ,\tilde{\psi }]$ with $\tilde{\psi }\in Z$ and $\alpha \in \mathbb {R}$. Thus,

$$\begin{aligned} L_{u_{\sigma _0}}\tilde{\psi }+\textrm{i}\alpha e=\tilde{\phi }\end{aligned}$$

(30)

and since $\textrm{i}e\perp _{L^2} \phi _{\sigma _0}^*$ by the definition of $\sigma _0$, the Fredholm alternative says that a necessary and sufficient condition for $\tilde{\phi }$ to satisfy (30) is that $\tilde{\phi }\in {{\,\textrm{span}\,}}\{\phi _{\sigma _0}^*\}^{\perp _{L^2}}$ as claimed. Note that in this case $\tilde{\psi }\in H^2_\text {per}(0,2\pi )=\ker L_{u_{\sigma _0}}\oplus Z$ and hence, for every given $\alpha \in \mathbb {R}$ and $\tilde{\phi }\in {{\,\textrm{span}\,}}\{\phi _{\sigma _0}^*\}^{\perp _{L^2}}$ there is a unique element $\tilde{\psi }\in Z$ that solves (30). $\square$

Proof of Theorem 3

The proof is divided into three steps.

Step 1. We begin by verifying for (26) the conditions for the local bifurcation theorem of Crandall–Rabinowitz, cf. Theorem 9. By Lemma 2, $\partial _{(f_1,v)} F(\sigma _0,0,0): \mathbb {R}\times Z\rightarrow L^2(0,2\pi )$ is an index-zero Fredholm operator and it satisfies

$$\begin{aligned} \ker \partial _{(f_1,v)} F(\sigma _0,0,0)={{\,\textrm{span}\,}}\{(1,\psi _{\sigma _0})\}, \end{aligned}$$

where $\psi _{\sigma _0}$ denotes the unique element of Z which solves (28). Hence, (H1) is satisfied. To see (H2) note that

$$\begin{aligned} \partial ^2_{(f_1,v),\sigma } F(\sigma _0,0,0)[1,\psi _{\sigma _0}]=2u_{\sigma _0}'\overline{u_{\sigma _0}}\psi _{\sigma _0}+2\overline{u_{\sigma _0}'} u_{\sigma _0} \psi _{\sigma _0}+2u_{\sigma _0}u_{\sigma _0}'\overline{\psi _{\sigma _0}}. \end{aligned}$$

On the other hand, differentiation of (28) w.r.t. s yields

$$\begin{aligned} L_{u_{\sigma _0}}\psi '_{\sigma _0} = 2u_{\sigma _0}'\overline{u_{\sigma _0}}\psi _{\sigma _0}+2\overline{u_{\sigma _0}'} u_{\sigma _0} \psi _{\sigma _0} +2u_{\sigma _0}u_{\sigma _0}'\overline{\psi _{\sigma _0}}-\textrm{i}e' \end{aligned}$$

(31)

so that

$$\begin{aligned} \partial ^2_{(f_1,v),\sigma } F(\sigma _0,0,0)[1,\psi _{\sigma _0}] = L_{u_{\sigma _0}}\psi _{\sigma _0}'+\textrm{i}e'. \end{aligned}$$

(32)

Hence, the characterization of ${{\,\textrm{range}\,}}\partial _{(f_1,v)} F(\sigma _0,0,0)$ from Lemma 2 implies that the transversality condition (H2) is satisfied if and only if ${\text {Re}}\int _0^{2\pi } \textrm{i}e'(s)\overline{\phi _{\sigma _0}^*(s)}\,ds \not =0$ which amounts to assumption (8). This already allows us to apply Theorem 9 and we obtain the existence of a local curve $t \mapsto (\sigma (t), f_1(t),v(t))$, $\dot{f_1}(0)=1$, $f_1(0)=0$, $v(0)=0$, $\sigma (0)=\sigma _0$ with $F(\sigma (t), f_1(t),v(t))=0$. Assertion (i) is then satisfied with $u(t) {:}{=}u_{\sigma (t)}+v(t)$. Assertion (ii) follows like in the proof of Theorem 2.

Step 2. From here on let us additionally assume that zero is an algebraically simple eigenvalue of $L_{u_0}$, i.e., $u_0' \notin {{\,\textrm{range}\,}}L_{u_0}$. Next we want to show that $L_{u(t)}$ is invertible for $0<|t|<\delta ^*$ and $\delta ^*$ sufficiently small, i.e., that the critical zero eigenvalue of $L_{u(0)}=L_{u_{\sigma _0}}$ moves away from zero when t evolves. Let us define

$$\begin{aligned} H:\left\{ \begin{array}{rcl} H^2_{\text {per}}(0,2\pi ) \times Z \times \mathbb {R}\ {} &{} \rightarrow &{} L^2(0,2\pi ), \\ (u,v,\mu ) &{} \mapsto &{} L_u (u_{\sigma _0}'+v) -\mu (u_{\sigma _0}'+v). \end{array} \right. \end{aligned}$$

Then, $H(u_{\sigma _0},0,0)=0$ and

$$\begin{aligned} \partial _{(v,\mu )} H (u_{\sigma _0},0,0):\left\{ \begin{array}{rcl} Z \times \mathbb {R}\ {} &{} \rightarrow &{} L^2(0,2\pi ), \\ (\psi ,\alpha ) &{} \mapsto &{} L_{u_{\sigma _0}} \psi -\alpha u_{\sigma _0}' \end{array} \right. \end{aligned}$$

clearly defines an isomorphism due to our assumption that $u_{\sigma _0}' \notin {{\,\textrm{range}\,}}L_{u_{\sigma _0}}$. By the implicit function theorem, we find neighborhoods $U \subset H^2_{\text {per}}(0,2\pi )$ of $u_{\sigma _0}$, $V\subset Z$ of 0, $J \subset \mathbb {R}$ of 0 and continuously differentiable functions $v^*:U \rightarrow V$, $\mu ^*:U \rightarrow J$ such that $v^*(u_{\sigma _0})=0$, $\mu ^*(u_{\sigma _0})=0$ and

$$\begin{aligned} \begin{aligned} \forall (u,v,\mu )\in U\times V\times J: \, H(u,v,\mu )=0 \quad \Longleftrightarrow \quad v=v^*(u), \mu =\mu ^*(u). \end{aligned} \end{aligned}$$

Thus, we find $L_{u(t)}\bigl (u_{\sigma _0}'+v^*(u(t))\bigr )=\mu ^*(u(t))\bigl (u_{\sigma _0}'+v^*(u(t))\bigr )$ for |t| sufficiently small. With $\varphi (t) {:}{=}u_{\sigma _0}'+v^*(u(t))$ and $\mu (t){:}{=}\mu ^*(u(t))$, we have $\varphi (0)=u_{\sigma _0}'$, $\mu (0)=0$ and

$$\begin{aligned} L_{u(t)} \varphi (t)=\mu (t)\varphi (t) \end{aligned}$$

(33)

so that we have found a parameterization of the eigenvalue $\mu (t)$ nearby 0 with eigenfunction $\varphi (t)$ of $L_{u(t)}$. Next we want to compute $\dot{\mu }(0)$ and show that $\dot{\mu }(0)\ne 0$ so that the critical zero eigenvalue moves away from zero. Differentiating (33) w.r.t. t and evaluating at $t=0$ we get

$$\begin{aligned} L_{u_{\sigma _0}} \dot{\varphi }(0)-2 \dot{u}(0) \overline{u_{\sigma _0}} u_{\sigma _0}' -2 u_{\sigma _0}\overline{\dot{u}(0)} u_{\sigma _0}'-2 u_{\sigma _0}\dot{u}(0) \overline{u_{\sigma _0}'}=\dot{\mu }(0) u_{\sigma _0}'. \end{aligned}$$

Theorem 9 yields $\dot{v}(0)=\psi _{\sigma _0}$ from which we find $\dot{u}(0)=-u_{\sigma _0}' \dot{\sigma }(0)+\psi _{\sigma _0}$. Thus,

$$\begin{aligned} L_{u_{\sigma _0}} \dot{\varphi }(0)-2(\psi _{\sigma _0}\overline{u_{\sigma _0}} u_{\sigma _0}'+ u_{\sigma _0}\overline{\psi _{\sigma _0}} u_{\sigma _0}'+ u_{\sigma _0}\psi _{\sigma _0}\overline{u_{\sigma _0}'})+2 \dot{\sigma }(0) u_{\sigma _0}'(\overline{u_{\sigma _0}} u_{\sigma _0}'+2u_{\sigma _0}\overline{u_{\sigma _0}'}) =\dot{\mu }(0) u_{\sigma _0}'. \end{aligned}$$

Using (31), this gives

$$\begin{aligned} L_{u_{\sigma _0}} \dot{\varphi }(0)-L_{u_{\sigma _0}} \psi _{\sigma _0}'-\textrm{i}e'+2\dot{\sigma }(0) u_{\sigma _0}'(\overline{u_{\sigma _0}} u_{\sigma _0}'+2u_{\sigma _0}\overline{u_{\sigma _0}'})=\dot{\mu }(0) u_{\sigma _0}'. \end{aligned}$$

Testing this equation with $\phi _{\sigma _0}^*$ and using $\dot{\mu }(0) \in \mathbb {R}$, we obtain

$$\begin{aligned} {\text {Re}}\int _0^{2\pi } -\textrm{i}e' \overline{\phi _{\sigma _0}^*}+2 \dot{\sigma }(0) u_{\sigma _0}'(\overline{u_{\sigma _0}} u_{\sigma _0}'+2u_{\sigma _0}\overline{u_{\sigma _0}'})\overline{\phi _{\sigma _0}^*} \, ds=\dot{\mu }(0) {\text {Re}}\int _0^{2\pi } u_{\sigma _0}'\overline{\phi _{\sigma _0}^*} \, ds. \end{aligned}$$

Further,

$$\begin{aligned} {\text {Re}}\int _0^{2\pi } u_{\sigma _0}'(\overline{u_{\sigma _0}} u_{\sigma _0}'+2u_{\sigma _0}\overline{u_{\sigma _0}'})\overline{\phi _{\sigma _0}^*} \, ds=0 \end{aligned}$$

as can be seen by differentiating (3) for $f_1=0$ twice w.r.t. s. Hence,

$$\begin{aligned} {\text {Re}}\int _0^{2\pi } -\textrm{i}e' \overline{\phi _{\sigma _0}^*} \, ds=\dot{\mu }(0) {\text {Re}}\int _0^{2\pi } u_{\sigma _0}'\overline{\phi _{\sigma _0}^*} \, ds. \end{aligned}$$

Due to $u_{\sigma _0}' \notin {{\,\textrm{range}\,}}L_{u_{\sigma _0}}$, we have ${\text {Re}}\int _0^{2\pi } u_{\sigma _0}'\overline{\phi _{\sigma _0}^*} \, ds \ne 0$ so that

$$\begin{aligned} \dot{\mu }(0)=\frac{{\text {Im}}\int \limits _0^{2\pi } e'(s+\sigma _0) \overline{\phi _0^*(s)} \, ds}{{\text {Re}}\int _0^{2\pi } u_0'\overline{\phi _0^*} \, ds} \end{aligned}$$

and the condition $\dot{\mu }(0)\ne 0$ amounts to assumption (8) of the theorem.

Finally, employing some arguments from spectral theory, we ensure that no other eigenvalue runs into zero. For $u=u_1+\textrm{i}u_2\in H^2_{\text {per}}(0,2\pi )$ let us define the $\mathbb {C}$-linear operator

$$\begin{aligned} L^{\mathbb {C}}_u:\left\{ \begin{array}{rcl} H^2_{\text {per}}((0,2\pi ),\mathbb {C}^2) \ {} &{} \rightarrow &{} L^2((0,2\pi ),\mathbb {C}^2) \\ \begin{pmatrix} \varphi _1 \\ \varphi _2 \end{pmatrix} &{} \mapsto &{} \begin{pmatrix} -d\varphi _1''-\omega \varphi _2'+\zeta \varphi _1+\varphi _2-3u_1^2 \varphi _1-u_2^2 \varphi _1-2u_1u_2\varphi _2 \\ -d \varphi _2''+\omega \varphi _1'+\zeta \varphi _2-\varphi _1-u_1^2\varphi _2-3u_2^2\varphi _2-2u_1u_2\varphi _1 \end{pmatrix} \end{array} \right. \end{aligned}$$

which is constructed in such a way that

$$\begin{aligned} L^{\mathbb {C}}_u \begin{pmatrix} \varphi _1 \\ \varphi _2 \end{pmatrix}= \begin{pmatrix} {\text {Re}}L_u(\varphi _1+\textrm{i}\varphi _2) \\ {\text {Im}}L_u(\varphi _1+\textrm{i}\varphi _2) \end{pmatrix} \end{aligned}$$

whenever $\varphi _1,\varphi _2\in H^2_{\text {per}}((0,2\pi ),\mathbb {R})$. Since $L^{\mathbb {C}}_u$ is an index-zero Fredholm operator, its spectrum consists of eigenvalues. The real part of these eigenvalues (weighted with ${{\,\textrm{sign}\,}}(d)$) is bounded from below by $c\in \mathbb {R}$ which is chosen such that

$$\begin{aligned} {\text {Re}}\biggl \langle {{\,\textrm{sign}\,}}(d) L_u^\mathbb {C}\begin{pmatrix} \varphi _1 \\ \varphi _2 \end{pmatrix},\begin{pmatrix} \varphi _1 \\ \varphi _2 \end{pmatrix} \biggr \rangle _{L^2((0,2\pi ),\mathbb {C}^2)} \ge c \left\| \begin{pmatrix} \varphi _1 \\ \varphi _2 \end{pmatrix} \right\| _{L^2((0,2\pi ),\mathbb {C}^2)}^2 \end{aligned}$$

holds. This implies that the resolvent set $\rho (L^{\mathbb {C}}_u)$ is non-empty, and the compactness of the embedding $H^2_{\text {per}}((0,2\pi ),\mathbb {C}^2) \hookrightarrow L^2((0,2\pi ),\mathbb {C}^2)$ ensures that $L^{\mathbb {C}}_u$ has compact resolvent so that $\sigma (L^{\mathbb {C}}_u)$ consists of isolated eigenvalues. Now choose $\varepsilon >0$ such that $\sigma (L_{u(0)}^\mathbb {C}) \cap \overline{B_{\varepsilon }^\mathbb {C}(0)}=\{0\}$. Using [15, Chapter Four, Theorem 3.18] we find that $\sigma (L^{\mathbb {C}}_{u(t)}) \cap B_{\varepsilon }^\mathbb {C}(0)$ exactly consists of one algebraically simple eigenvalue if |t| is sufficiently small. If in addition |t| is chosen so small that $\mu (t) \in (-\varepsilon ,\varepsilon )$ then this means $\sigma (L^{\mathbb {C}}_{u(t)}) \cap B_{\varepsilon }^\mathbb {C}(0)=\{\mu (t)\}$. But from $\dot{\mu }(0)\ne 0$, we know that $\mu (t)\ne 0$ for small $|t|> 0$ which guarantees that $0\notin \sigma (L_{u(t)}^\mathbb {C})$ for $0<|t|<\delta ^*$ and $\delta ^*$ sufficiently small. Finally, $L_{u(t)}$ inherits the invertibility of $L_{u(t)}^\mathbb {C}$.

Step 3. Due to $\dot{f_1}(0)=1$ and Step 2, there exists a local reparameterization $({\tilde{f}}_1,u({\tilde{f}}_1))$ of $C(t)=(f_1(t),u(t))$ such that $L_{u({\tilde{f}}_1)}$ is invertible for $0<{\tilde{f}}_1<f_1^*$. Next we construct the connected set ${\mathcal {C}}^+_*$. For this we want to apply Theorem 8 to the map $T:\mathbb {R}\times H^1_{\text {per}}(0,2\pi ) \rightarrow H^1_{\text {per}}(0,2\pi )$ from the proof of Theorem 2. Note that this theorem cannot be applied directly at the point $(0,u_{\sigma _0})$ since $\partial _u T(0,u_{\sigma _0})$ is not invertible. Instead, we apply it to the points $({\tilde{f}}_1,u({\tilde{f}}_1))$ with ${\tilde{f}}_1\in (0,f_1^*)$ and obtain that the maximal continuum ${\mathcal {C}}^+(\tilde{f}_1)\subset [{\tilde{f}}_1,\infty )\times H^1_{\text {per}}(0,2\pi )$ of solutions of (3) with $(\tilde{f}_1, u({\tilde{f}}_1)) \in {\mathcal {C}}^+({\tilde{f}}_1)$ is unbounded or returns to another solution $u^+({\tilde{f}}_1)\ne u({\tilde{f}}_1)$ at $f_1={\tilde{f}}_1$. As in the proof of Theorem 2, we see that the continuum ${\mathcal {C}}^+({\tilde{f}}_1)$ persists as a connected and closed set in $[{\tilde{f}}_1,\infty )\times H^2_{\text {per}}(0,2\pi )$. Let us define

$$\begin{aligned} {\mathcal {C}}_*^+ {:}{=}\bigcup _{{\tilde{f}}_1 \in (0,f_1^*)} {\mathcal {C}}^+({\tilde{f}}_1)\subset {\mathcal {C}}^+. \end{aligned}$$

Clearly, ${\text {pr}}_1({\mathcal {C}}^{+}_*)\subset (0,\infty )$ and ${\mathcal {C}}^+_*$ is connected since ${\mathcal {C}}^+({\tilde{f}}_1) \subset {\mathcal {C}}^+({\bar{f}}_1)$ for ${\bar{f}}_1<{\tilde{f}}_1$. Let us now suppose that ${\text {pr}}_1(\mathcal {C^+_*})\ne (0,\infty )$ so that ${\text {pr}}_1({\mathcal {C}}^+_*)$ is bounded. By (ii), this implies that ${\mathcal {C}}^+_*$ is bounded too. Hence, ${\mathcal {C}}^+({\tilde{f}}_1)$ is bounded for ${\tilde{f}}_1 \in (0,f_1^*)$ and contains the additional element $(\tilde{f}_1,u^+({\tilde{f}}_1))$. Let us take ${\tilde{f}}_1= \frac{1}{n}$ and consider the two sequences of solutions $(\frac{1}{n}, u(\frac{1}{n}))_{n}$ and $(\frac{1}{n}, u^+(\frac{1}{n}))_{n}$. Using Theorem 5, we obtain uniform $C^3$-bounds for both sequences $(u(\frac{1}{n}))_{n}$ and $(u^+(\frac{1}{n}))_{n}$. Therefore, we can take convergent subsequences (denoted by the same index) and obtain $u(\frac{1}{n})\rightarrow u_{\sigma _0}$ and $u^+(\frac{1}{n})\rightarrow u_0^+$ in $C^2([0,2\pi ])$ as $n\rightarrow \infty$. In particular $(0,u_{\sigma _0}), (0,u_0^+)\in \overline{\mathcal {C^+_*}}$ and the uniqueness property from (i) guarantees that $u_0^+\ne u_{\sigma _0}$. This finishes the proof. $\square$

Proof of Corollary 2

We first check assumption (7) of Theorem 3. For $e(s)=\textrm{e}^{\textrm{i}k_1 s}$ we have

$$\begin{aligned} {\text {Im}}\int \limits _0^{2\pi } e(s+\sigma _0) \overline{\phi _0^*(s)} \,ds&= {\text {Im}}\int _0^{2\pi } \textrm{e}^{\textrm{i}k_1 (s+\sigma _0)} \overline{\phi _0^*(s)} \,ds \\&= \cos (k_1\sigma _0) {\text {Im}}\int \limits _0^{2\pi } \textrm{e}^{\textrm{i}k_1 s} \overline{\phi _0^*(s)} \, ds+\sin (k_1\sigma _0){\text {Re}}\int _0^{2\pi } \textrm{e}^{\textrm{i}k_1 s} \overline{\phi _0^*(s)} \, ds, \end{aligned}$$

where

$$\begin{aligned} {\text {Im}}\int \limits _0^{2\pi } \textrm{e}^{\textrm{i}k_1 s} \overline{\phi _0^*(s)} \, ds&= \int _0^{2\pi } \sin (k_1 s) {\text {Re}}\phi _0^*(s)-\cos (k_1 s){\text {Im}}\phi _0^*(s) \, ds, \\ {\text {Re}}\int _0^{2\pi } \textrm{e}^{\textrm{i}k_1 s} \overline{\phi _0^*(s)} \, ds&=\int _0^{2\pi } \cos (k_1 s) {\text {Re}}\phi _0^*(s) +\sin (k_1 s) {\text {Im}}\phi _0^*(s) \, ds. \end{aligned}$$

Since assumption (9) guarantees that ${\text {Im}}\int _0^{2\pi } \textrm{e}^{\textrm{i}k_1 s} \overline{\phi _0^*(s)} \, ds$ and ${\text {Re}}\int _0^{2\pi } \textrm{e}^{\textrm{i}k_1 s} \overline{\phi _0^*(s)} \, ds$ do not vanish simultaneously condition (10) ensures that assumption (7) of Theorem 3 is fulfilled.

Next we check that assumption (8) of Theorem 3 holds. For this we compute

$$\begin{aligned} {\text {Im}}\int \limits _0^{2\pi }e'(s+\sigma _0)\overline{\phi _0^*(s)}\,ds ={\text {Im}}\int \limits _0^{2\pi }\textrm{i}k_1 \textrm{e}^{\textrm{i}k_1(s+\sigma _0)}\overline{\phi _0^*(s)}\,ds = k_1 {\text {Re}}\int _0^{2\pi }\textrm{e}^{\textrm{i}k_1(s+\sigma _0)}\overline{\phi _0^*(s)}\,ds. \end{aligned}$$

(34)

From (9) we know that $\int _0^{2\pi }\textrm{e}^{\textrm{i}k_1(s+\sigma _0)}\overline{\phi _0^*(s)}\,ds = \textrm{e}^{\textrm{i}k_1\sigma _0} \int _0^{2\pi }\textrm{e}^{\textrm{i}k_1s}\overline{\phi _0^*(s)}\,ds \ne 0$ and, moreover, we see that ${\text {Im}}\int _0^{2\pi }\textrm{e}^{\textrm{i}k_1(s+\sigma _0)}\overline{\phi _0^*(s)}\,ds=0$ by the definition of $\sigma _0$. Therefore, the expression in (34) does not vanish and so assumption (8) of Theorem 3 holds. This is all we had to show. $\square$

Proof of Theorem 4

Let us fix all parameters $d, \omega , \zeta , k_1$ and $f_0$ and consider $u: f_1\mapsto u(f_1)$ as a function mapping the parameter $f_1\in [-f_1^*,f_1^*]$ to the uniquely defined solution of (2) in the neighborhood of the trivial solution $u_0$. The existence of such a smooth function follows from the implicit function theorem applied to the equation $T(f_1,u)=0$, cf. proof of Theorem 2. Similarly we consider the functions $v: f_1\mapsto \frac{d u(f_1)}{df_1}$ and $w: f_1\mapsto \frac{d^2 u(f_1)}{df_1^2}$. Then,

$$\begin{aligned} \frac{d}{df_1} \Vert u(f_1)\Vert _2^2 = 2\int _0^{2\pi } {\text {Re}}(u\overline{v})\,ds, \qquad \frac{d^2}{df_1^2} \Vert u(f_1)\Vert _2^2 = 2\int _0^{2\pi } {\text {Re}}(u{\overline{w}})+|v|^2\,ds \end{aligned}$$

(35)

and the differential equations for v, w at $f_1=0$ are given by

$$\begin{aligned} -d v'' + \textrm{i}\omega v' + (\zeta -\textrm{i})v - 2|u_0|^2 v-u_0^2\overline{v}+\textrm{i}\textrm{e}^{\textrm{i}k_1 s}= & {} 0, \end{aligned}$$

(36)

$$\begin{aligned} -d w'' + \textrm{i}\omega w' + (\zeta -\textrm{i})w - 4u_0|v|^2-2\overline{u_0} v^2-2|u_0|^2 w-u_0^2\overline{w}= & {} 0 \end{aligned}$$

(37)

both equipped with $2\pi$-periodic boundary conditions. The first equation (36) has a unique solution since the homogeneous equation has a trivial kernel, cf. proof of Theorem 2. Thus, $v(s)=\alpha \textrm{e}^{\textrm{i}k_1 s}+\beta \textrm{e}^{-\textrm{i}k_1s}$ where $\alpha , \beta \in \mathbb {C}$ solve the linear system

$$\begin{aligned} (dk_{1}^2-k_{1}\omega +\zeta -\textrm{i}-2|u_0|^2)\alpha - u_0^2\overline{\beta }+\textrm{i}&= 0,\\ (dk_{1}^2+k_{1}\omega +\zeta -\textrm{i}-2|u_0|^2)\beta - u_0^2\overline{\alpha }&=0. \end{aligned}$$

Solving for $\alpha , \beta$ leads to the formulae in the statement of the theorem. Since v is the sum of two $2\pi$-periodic complex exponentials and $u_0$ is a constant we see from (35) that $\frac{d}{df_1} \Vert u(f_1)\Vert _2^2\mid _{f_1=0} =0$. Having determined v, we can consider the second equation (37) as an inhomogeneous equation for w. It also has a unique solution since the homogeneous equation is the same as in (36). Since the inhomogeneity is of the form $c_1 \textrm{e}^{\textrm{i}2k_1s}+c_2 \textrm{e}^{-\textrm{i}2k_1 s}+c_3$ the solution has the form $w(s) = \gamma \textrm{e}^{\textrm{i}2k_1 s}+\delta \textrm{e}^{-\textrm{i}2 k_1s}+\epsilon$. Moreover, for the determination of $\frac{d^2}{df_1^2} \Vert u(f_1)\Vert _2^2$ the values of $\gamma , \delta$ are irrelevant and only the value of $\epsilon$ matters. Using

$$\begin{aligned} |v|^2 = |\alpha |^2+|\beta |^2+2{\text {Re}}(\alpha \overline{\beta }\textrm{e}^{\textrm{i}2 k_1 s}), \quad v^2 = \alpha ^2 \textrm{e}^{\textrm{i}2k_1 s}+\beta ^2 \textrm{e}^{-\textrm{i}2k_1 s}+2\alpha \beta \end{aligned}$$

we find from (37) that the equation determining $\epsilon$ is

$$\begin{aligned} (\zeta -\textrm{i})\epsilon -4u_0(|\alpha |^2+|\beta |^2)-4\overline{u_0} \alpha \beta -2 |u_0|^2\epsilon -u_0^2 \overline{\epsilon }=0. \end{aligned}$$

Since this is an equation of the form $x\epsilon + y\overline{\epsilon }=z$ with x, y, z given in the statement of the theorem we find the solution formula $\epsilon =\frac{-{\overline{z}} y+z\overline{x}}{|x|^2-|y|^2}$. Finally, only the constant contributions from ${\overline{w}}$ and $|v|^2$ contribute to the integral in the formula (35) for $\frac{d^2}{df_1^2} \Vert u(f_1)\Vert _2^2$ and lead to the claimed statement of the theorem. $\square$

Appendix

Here, we raise the issue mentioned in Remark 5.($\gamma$) that assumption (9) from Corollary 2 is not satisfied if $u_0$ is $\tfrac{2\pi }{j}$-periodic and $j\in \mathbb {N}$ is not a divisor of $k_1$. Let us first prove that $\phi _0^*$ (spanning $\ker L_{u_0}^*$) inherits several properties from $u_0'$ (spanning $\ker L_{u_0}$).

Proposition 2

Let $u_0\in H^2_{\text {per}}(0,2\pi )$ be a non-constant non-degenerate solution of (3) for $f_1=0$ and let $\ker L^*_{u_0}={{\,\textrm{span}\,}}\{\phi _0^*\}$. Then, the following holds:

• (i) If $u_0$ is $\frac{2\pi }{j}$-periodic with $j\in \mathbb {N}$ then $\phi _0^*$ is $\frac{2\pi }{j}$-periodic.

• (ii) If $\omega =0$ and if $u_0$ is even then $\phi _0^*$ is odd.

Proof

(i) By assumption we have that $\ker L_{u_0} = {{\,\textrm{span}\,}}\{u_0'\}$ and $u_0'$ is a $\frac{2\pi }{j}$-periodic function. Let us define $D {:}{=}\{\varphi \in H_{\text{ per }}^2(0,2\pi ):\varphi \text{ is } \frac{2\pi }{j} -\text{ periodic }\}$ and similarly $L^2_{j}(0,2\pi )=\{\varphi \in L^2(0,2\pi ):\varphi \text { is } \frac{2\pi }{j}-\text {periodic}\}$. If we consider the restriction

$$\begin{aligned} L_{u_0}^\#:\left\{ \begin{array}{rcl} D &{} \rightarrow &{} L^2_j(0,2\pi ), \\ \varphi &{} \mapsto &{} L_{u_0} \varphi , \end{array} \right. \end{aligned}$$

then $L_{u_0}^\#$ is again an index-zero Fredholm operator with $\ker L_{u_0}^\#= {{\,\textrm{span}\,}}\{u_0'\}$. Further we have $(L_{u_0}^\#)^* = (L_{u_0}^*)^\#$ where

$$\begin{aligned} (L_{u_0}^*)^\#:\left\{ \begin{array}{rcl} D &{} \rightarrow &{} L^2_j(0,2\pi ), \\ \varphi &{} \mapsto &{} L_{u_0}^* \varphi \end{array} \right. \end{aligned}$$

is the restriction of the adjoint. But since $1=\dim \ker (L_{u_0}^*)^\# = \dim \ker L_{u_0}^*$ it follows that $\ker (L_{u_0}^*)^\# = \ker L_{u_0}^*$ and hence $\phi _0^* \in D$ as claimed.

The proof of (ii) is very similar. Due to the assumption $\omega =0$, we can restrict both the domain and the codomain of $L_{u_0}$ to odd functions and observe that it is still an index-zero Fredholm operator. $\square$

Instead of $k_1\in \mathbb {N}$, let us now consider a perturbation $k_1(\epsilon )\in \mathbb {R}{\setminus }\{k_1\}$ with $\lim _{\epsilon \rightarrow 0} k_1(\epsilon )=k_1$. For $\epsilon \approx 0$, one may have maximally connected continua ${{\mathcal {C}}}_\epsilon ^+$ as described in Theorem 3. In a topological sense, one can describe $\liminf \{\mathcal C_{\epsilon }^+:\epsilon ^{-1}\in \mathbb {N}\}$ and $\limsup \{\mathcal C_{\epsilon }^+:\epsilon ^{-1}\in \mathbb {N}\}$ as in [31]. However, having in mind sequences of loops degenerating to one point, we do not intend to make any existence statement about a bifurcating branch obtained through such a topological limiting procedure. Let us abbreviate by $e_{\epsilon }(s)$ the periodic extension of $[0,2\pi )\rightarrow \mathbb {C}, \, s\mapsto \textrm{e}^{\textrm{i}k_1(\epsilon )s}$ onto $\mathbb {R}$. Note that

$$\begin{aligned} {\text {Im}}\int \limits _0^{2\pi } e_{\epsilon } (s+\sigma _{0,\epsilon }) \overline{\phi _0^*(s)} \, ds =&{\text {Im}}\int \limits _0^{2\pi } \textrm{e}^{\textrm{i}k_1(\epsilon )s} \overline{\phi _{\sigma _{0,\epsilon }}^*(s)} \, ds \\ =&{\text {Im}}\int \limits _{-\sigma _{0,\epsilon }}^{2\pi -\sigma _{0,\epsilon }} \textrm{e}^{\textrm{i}k_1(\epsilon )(s+\sigma _{0,\epsilon })} \overline{\phi _0^*(s)} \, ds \\ =&\cos (k_1(\epsilon )\sigma _{0,\epsilon }) {\text {Im}}\int \limits _{-\sigma _{0,\epsilon }}^{2\pi -\sigma _{0,\epsilon }} \textrm{e}^{\textrm{i}k_1(\epsilon )s} \overline{\phi _0^*(s)} \, ds\\&+\sin (k_1(\epsilon )\sigma _{0,\epsilon }) {\text {Re}}\int _{-\sigma _{0,\epsilon }}^{2\pi -\sigma _{0,\epsilon }} \textrm{e}^{\textrm{i}k_1(\epsilon )s} \overline{\phi _0^*(s)} \, ds \end{aligned}$$

so that assumption (7) from Theorem 3 becomes

$$\begin{aligned} \tan (k_1(\epsilon )\sigma _{0,\epsilon }) =\frac{\int _{-\sigma _{0,\epsilon }}^{2\pi -\sigma _{0,\epsilon }} \cos (k_1(\epsilon )s) {\text {Im}}\phi _0^*(s) -\sin (k_1(\epsilon )s) {\text {Re}}\phi _0^*(s) \, ds}{\int _{-\sigma _{0,\epsilon }}^{2\pi -\sigma _{0,\epsilon }} \sin (k_1(\epsilon )s) {\text {Im}}\phi _0^*(s)+ \cos (k_1(\epsilon )s) {\text {Re}}\phi _0^*(s) \, ds}. \end{aligned}$$

One may expect that if (as a result of such a limiting procedure) a bifurcating branch at $k_1=\lim _{\epsilon \rightarrow 0} k_1(\epsilon )$ exists then it bifurcates at $\sigma _0=\lim _{\epsilon \rightarrow 0} \sigma _{0,\epsilon }$ determined from

$$\begin{aligned} \tan (k_1\sigma _0)&= \lim _{\epsilon \rightarrow 0} \frac{\int _{-\sigma _{0,\epsilon }}^{2\pi -\sigma _{0,\epsilon }} \cos (k_1(\epsilon )s) {\text {Im}}\phi _0^*(s) -\sin (k_1(\epsilon )s) {\text {Re}}\phi _0^*(s) \, ds}{\int _{-\sigma _{0,\epsilon }}^{2\pi -\sigma _{0,\epsilon }} \sin (k_1(\epsilon )s) {\text {Im}}\phi _0^*(s)+ \cos (k_1(\epsilon )s) {\text {Re}}\phi _0^*(s) \, ds}\\&=\frac{\int _{-\sigma _{0}}^{2\pi -\sigma _{0}} s\sin (k_1 s){\text {Im}}\phi _0^*(s)+s \cos (k_1 s){\text {Re}}\phi _0^*(s)\,ds}{\int _{-\sigma _{0}}^{2\pi -\sigma _{0}} s\sin (k_1 s){\text {Re}}\phi _0^*(s)-s \cos (k_1 s){\text {Im}}\phi _0^*(s)\,ds}. \end{aligned}$$

However, this is not supported by our numerical experiments and we have to leave the correct determination of $\sigma _0$ in this case as an open question.