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

We consider Kerr frequency combs in a dual-pumped microresonator as time-periodic and spatially 2π\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$2\pi $$\end{document}-periodic traveling wave solutions of a variant of the Lugiato–Lefever equation, which is a damped, detuned and driven nonlinear Schrödinger equation given by iaτ=(ζ-i)a-daxx-|a|2a+if0+if1ei(k1x-ν1τ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textrm{i}a_\tau =(\zeta -\textrm{i})a-d a_{x x}-|a|^2a+\textrm{i}f_0+\textrm{i}f_1\textrm{e}^{\textrm{i}(k_1 x-\nu _1 \tau )}$$\end{document}. The main new feature of the problem is the specific form of the source term f0+f1ei(k1x-ν1τ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f_0+f_1\textrm{e}^{\textrm{i}(k_1 x-\nu _1 \tau )}$$\end{document} which describes the simultaneous pumping of two different modes with mode indices k0=0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k_0=0$$\end{document} and k1∈N\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k_1\in \mathbb {N}$$\end{document}. 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., f1=0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f_1=0$$\end{document}, can be continued into the range f1≠0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f_1\not =0$$\end{document}. Our analytical findings apply both for anomalous (d>0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$d>0$$\end{document}) and normal (d<0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$d<0$$\end{document}) dispersion, and they are illustrated by numerical simulations.


INTRODUCTION
Optical frequency comb devices are extremely promising in many applications such as, e.g., optical frequency metrology [25], spectroscopy [20,27], ultrafast optical ranging [24], and high capacity optical communications [14].For many of these applications the Kerr soliton combs are generated by using a monochromatic pump.However, recently new pump schemes have been discussed, where more than one resonator mode is pumped, cf.[23].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.[7] 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 [9,12,16].Our analysis relies on a variant of the LLE which is modified for two-mode pumping, cf.[23] and [7] for a derivation.Using dimensionless, normalized quantities this equation takes the form Here, a(τ, x) represents the optical intracavity field as a function of normalized time τ = κ 2 t and angular position x ∈ [0, 2π] within the ring resonator.The constant κ > 0 describes the cavity decay rate and d = 2 κ d 2 quantifies the dispersion in the system (where ω k = ω 0 + d 1 k + d 2 k 2 is the cavity dispersion relation between the resonant frequencies ω k and the relative indices k ∈ 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 ∈ Z with k 0 = 0 being the first and k 1 ∈ N the second pumped mode.With f 0 , f 1 we describe the normalized power of the two input pumps and ω p 0 , ω 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 ζ = 2 κ (ω 0 − ω p 0 ) and ζ 1 = 2 κ (ω k 1 − ω p 1 ).They describe the offsets of the input pump frequencies ω p 0 and ω p 1 to the closest resonance frequency ω 0 and ω k 1 of the microresonator.The particular form of the pump term i f 0 + i f 1 e i(k 1 x−ν 1 τ) with ν 1 = ζ − ζ 1 + dk 2  1 suggests to change into a moving coordinate frame and to study solutions of (1) of the form a(τ, x) = u(s) with s = x − ωτ and ω = ν 1 k 1 .These traveling wave solutions propagate with speed ω in the resonator and their profiles u solve the ordinary differential equation (2) − du + iωu + (ζ − i)u − |u| 2 u + i f 0 + i f 1 e ik 1 s = 0, u 2π-periodic.
In the case f 1 = 0 equation (1) amounts to the case of pumping only one mode.This case has been thoroughly studied, e.g. in [5,6,8,9,13,15,16,17,18,19,22].In this paper we are interested in the case f 1 = 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 with a 2π-periodic (not necessarily continuous) function e : R → C and f 0 , f 1 ∈ R. Hence, we consider the LLE (3) − du + iωu + (ζ − i)u − |u| 2 u + i f (s) = 0, u 2π-periodic.
Our main results on the existence of solutions to (3) are stated in Section 2. In Section 3 we illustrate our main analytical results by numerical simulations.The proofs of the main results are given in Section 4 (a-priori bounds), Section 5 (existence and uniqueness), and Section 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.

MAIN RESULTS
In the following we state our main results.
• Theorem 1 provides existence of at least one solution of (3) for any choice of the parameters and any choice of f .• Theorem 6 and Corollary 8 describe how trivial (constant) solutions from the special case f 1 = 0 can be continued into non-trivial solutions for f 1 = 0. • Theorem 9 and Corollary 10 show how a non-trivial solution from the case f 1 = 0 can be continued to f 1 = 0. 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.A corresponding uniqueness result, which applies whenever |ζ| 1 is sufficiently large or (essentially) f 2 1 is sufficiently small is given in Theorem 17 in Section 5 together with more precise details.
We will use the following Sobolev spaces.For k ∈ N the space H k (0, 2π) consists of all square-integrable functions on (0, 2π) whose weak derivatives up to order k exist and are square-integrable on (0, 2π).By H k per (0, 2π) we denote all locally square-integrable 2πperiodic functions on R whose weak derivatives up to order k exist and are locally squareintegrable on R. In both spaces the norm is given by u = ∑ k j=0 ( d ds ) j u 2 L 2 (0,2π) Clearly H k per (0, 2π) is a proper subspace of H k (0, 2π) since u ∈ H k per (0, 2π) implies that ( d ds ) j u(0) = ( d ds ) j u(2π) for j = 0, . . ., k − 1.Unless otherwise stated, all of the above Hilbert spaces are spaces of complex valued functions over the field R. In particular, for v, w ∈ L 2 (0, 2π) we use the inner product v, w 2 := Re 2π 0 vw ds.The induced norm is denoted by • 2 .
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 = 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 Section 2.3.

2.1.
One-sided continuation of trivial solutions.In the special case f 1 = 0 there are trivial (constant) solutions u 0 ∈ C of (3) satisfying the algebraic equation ( 4) From [13, Lemma 2.1] we know that for given f 0 ∈ R the curve of constant solutions can be parameterized by (5) In Figure 1 we show the curve of the squared L 2 -norm of all constant solutions of (3) for and f 0 = 2.The curve may or may not have turning points which are characterized by ζ (t) = 0.This condition can be formulated independently of t by the equivalent condition By a straightforward analysis one can show that with we have • exactly one (degenerate) turning point for • exactly two turning points for Starting from f 1 = 0 we use a kind of global implicit function theorem to continue a constant solution u 0 ∈ C of (3) with respect to f 1 .This procedure is analyzed in Theorem 6.The continuation works if the constant solution u 0 ∈ C is non-degenerate in the following sense.
) and hence an index-zero Fredholm operator.Notice also that span{u } always belongs to the kernel of L u .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 constant solution u 0 is non-degenerate if the linearized operator L u 0 is injective, and, as a consequence, invertible in suitable spaces.

Lemma 4.
A trivial solution u 0 ∈ C of (3) for f 1 = 0 is non-degenerate if and only if (a) Case ω = 0: Proof.Let ϕ ∈ H 2 per (0, 2π) be in the kernel of the linearized operator, i.e., −dϕ This implies that the Fourier coefficients ϕ m of the Fourier series ϕ = ∑ m∈Z ϕ m e ims have the property that If we also write down the complex conjugate of this equation then we see that non-degeneracy of u 0 is equivalent to the non-vanishing of the determinant for this two-by-two system in the variables ϕ m , ϕ −m for all m ∈ N 0 .Computing the determinant we obtain the condition In the case ω = 0 this is trivially satisfied for all m = 0 (because then the imaginary part is non-zero) and for m = 0 by assumption (a) of the lemma.In the case ω = 0 condition (6) can only be guaranteed by assumption (b).
Remark 5. Trivial solutions of (3) for f 1 = 0 are determined by (4).For ω = 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 ω = 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.ζ from the curve of trivial solutions is fulfilled if and only if the expression in (b) vanishes for at least one m ∈ N, cf.[6], [13].
per (0, 2π) is bounded for any M > 0.Moreover, if pr 1 (C + ) denotes the projection of C + onto the f 1 -parameter component, then at least one of the following properties hold: per (0, 2π) with corresponding properties also exists.

Remark 7.
If property (a) of Theorem 6 holds, then C + is unbounded in the direction of the parameter f 1 ∈ [0, ∞) and hence this is an existence result for all f 1 ∈ [0, ∞).Property (b) means that the continuum C + returns to the f 1 = 0 line at a point u + 0 = u 0 .Corollary 8. Property (a) in Theorem 6 holds in any of the following three cases, * A continuum is a closed and connected set. where 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 ω = 0 there is a plethora of non-trivial solutions, cf.[6], [13].For ω = 0 we do not know whether non-trivial solutions exist for f 1 = 0.The fact that for ω = 0 there are no bifurcations from the curve of trivial solutions indicates that there may be no solutions other than the trivial ones.Although by the current state of understanding the hypotheses of Theorem 9 (see below) can only be fulfilled for ω = 0, we allow in the following for general ω ∈ 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 σ (s) := u 0 (s − σ), then u σ also solves (3) for f 1 = 0. Hence
For the special choice e(s) = e ik 1 s Theorem 9 takes the following form.(10) in Corollary 10 guarantees that the numerator and the denominator of the right-hand side of (11) do not vanish simultaneously.In the case where the denominator vanishes, Equation ( 11) is to be read as cos(k 1 σ 0 ) = 0.In the interval [0, π k 1 ) equation ( 11) has a unique solution σ 0 ∈ [0, π k 1 ).All solutions of (11) in [0, 2π) are then given by σ 0 + j π k 1 for j = 0, . . ., 2k 1 − 1.This can result in up to 2k 1 bifurcation points.Smaller periodicities of u 0 may reduce the actual number of different bifurcation points.E.g., if k 1 ≥ 2 and if u 0 has smallest period 2π k 1 then only two bifurcation points exist.(γ) Let j ∈ N not be a divisor of k 1 and u 0 be 2π j -periodic.Then assumption (10) is not satisfied since φ * 0 inherits the periodicity of u 0 .We will say more about this case in the Appendix.
(δ) The non-trivial solutions u 0 of (3) for f 1 = 0 and ω = 0 constructed in [6], [13] are even around s = 0.In this case, (9) is not an additional assumption because it coincides with assumption (8).The reason is that φ * 0 (spanning ker L * u 0 ) inherits the parity of u 0 (spanning ker L u 0 ) which implies 2π 0 u 0 u 0 + 2u 0 u 0 u 0 φ * 0 ds = 0, cf.Proposition 22. Also, the value of σ 0 in Corollary 10 is determined by the simpler expression It is an open problem if (3) admits solutions for f 1 = 0 and ω = 0 which (up to a shift) are not even around s = 0. ( ) Note that in property (b) we exclude that u + 0 = u σ 0 but we do not exclude that u + 0 coincides with a shift of u 0 different from u σ 0 .

Two-sided continuations.
Here we explain how we can use the results of Theorem 6 and Theorem 9, Corollary 10 for the continua C + and 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 Theorem 6 and Theorem 9: Next we assume that the generalized forcing term f (s) = f 0 + f 1 e(s) satisfies the symmetry condition that e s + π k 1 = −e(s) for some k 1 ∈ N.This symmetry condition is motivated by ( 2) where e(s) = e ik 1 s .If we denote by R the reflection operator which acts on solution pairs and is given by then, again both for the setting of Theorem 6 and Theorem 9, the continuum C has the following property: 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.
As another consequence, we have that either pr 1 (C) = (−∞, ∞) or pr 1 (C) is bounded from above and below.In the latter case, we call C a loop.
Our final result builds upon Theorem 6 and the resulting two-sided continuation of a trivial solution u 0 .It describes the shape of the L 2 -projection of the continuum C locally near (0, u 0 ).In particular, local convexity or concavity can be read from this result.In Section 3 we will put this result into perspective with numerical simulations of the f 1continuation of trivial solutions.
Theorem 13.Assume that the assumptions of Theorem 6 are satisfied and that additionally e(s) = e ik 1 s is fixed for a k 1 ∈ N. Then we can determine the local shape of the curve f 1 → u( f 1 ) 2  2 as follows:

NUMERICAL ILLUSTRATION OF THE ANALYTICAL RESULTS
In this section we restrict ourselves to equation (2), i.e., we fix e(s) = e ik 1 s .For this choice, we know from Section 2.3 that the one-sided continua C + and C − are related by C − = R(C + ).The following numerical examples were computed with d = −0.1, Figure 2 illustrates some of the two-sided continua C + ∪ C − obtained by continuation of trivial solutions for different values of the detuning ζ.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 Figure 2, the quantity 1 2π u 2 2 was used for this purpose.The black curve corresponds to spatially constant solutions of (2) obtained for f 1 = 0 and ζ ∈ [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 The fact that two functions have (nearly) the same norm does, of course, not imply that the functions themselves are (nearly) identical.It can be checked, however, that the two solutions which correspond to the two points where the distance between the two continua is minimal are indeed very similar (data not shown).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 13 shows that this is not true.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 13 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 → u( f 1 ).The type of local extremum is described by the sign of the second derivative 5) we can illustrate the signchanges of the second derivative.In Figure 5 we are plotting the curve t → (ζ(t), |u 0 (t)| 2 ) and indicate at each point on the curve the sign of 4π(Re(u 0 (t) , where (t), α(t), β(t) are taken from Theorem 13 with ζ = ζ(t) and u 0 = u 0 (t).In this particular example, as we run through the curve of trivial solutions from left to right a first sign-change of Recall that for ω = 0 there is a plethora of non-trivial solutions of (2) for f 1 = 0, cf.[6], [13].In fact, this time we find additional primary and secondary bifurcation branches for f 1 = 0 which are illustrated in Figure 6   At ζ = 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 ζ = 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 grey 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 eightshaped 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 6 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 9 applied to the nonconstant 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 2π.Thus, according to Remark 11.(β) exactly two shifts of it, which differ by π, 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 π) in the point where the orange lines intersect the grey line of non-trivial solutions.The two shifts cannot be distinguished in the picture, because a shift does not change the L 2 -norm.To illustrate the different continua for ζ = 3.6, we provide a zoom in Figure 7.We obtain again an unbounded continuum and a figure eight continuum.However, here we also FIGURE 7. Zoom at ζ = 3.6.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 ≥ 0 are described by case (b) of Theorem 9 applied to one of the non-trivial solutions u 0 at f 1 = 0 on it.They have no smaller period than 2π (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 grey line at exactly two points.The light blue curve does the same, but at π-shifts of these points.
For ζ = 3.3 the situation is more complicated.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 π-periodic while the remaining ones have smallest period 2π.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 2π this is a direct consequence of Theorem 9, cf.Remark 11.(β).However, at the three remaining π-periodic solutions at f 1 = 0 Theorem 9 does not apply, cf.Remark 11.(γ).Nevertheless, we observe continuations from these points.Interestingly, these points seem to be characterized by horizontal tangents, at least in this example.
For ζ = 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 Figure 9, where we also depict several of the continued solutions u of (2) for f 1 = 0. Since u is complex-valued, we use the quantity |u(s)| 2 for illustration purposes  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 (π-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 non-zero shift.Continuations from π-periodic solutions at f 1 = 0 are not covered by Theorem 9. 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.

PROOF OF A-PRIORI BOUNDS
We use the notation r + = max{0, r} to denote the positive part of any real number r ∈ R and also 1 d<0 to denote (as a function of d ∈ R) the characteristic function of the interval (−∞, 0).We write • p for the standard norm on L p (0, 2π) for p ∈ [1, ∞].A continuous map between two Banach spaces is said to be compact if it maps bounded sets into relatively compact sets.
hold, where For ζ sign(d) −C 2 1 d<0 these bounds can be improved to where . Remark 15.The improvement in the second part of the theorem lies in the fact that the bound D becomes small when the detuning ζ is such that ζ sign(d) is very negative.
Proof.The proof is divided into five steps.
Step 1.We first prove the L 2 estimate (15) To this end we multiply the differential equation ( 3) with ū to obtain ( 16) Taking the imaginary part yields Then H = h by equation ( 17) and H(0) = H(2π) by the periodicity of u.Hence Step 2. Next we prove ( 18) 2 ≤ BF From (3) we may isolate the linear term u and insert its derivative u into the following calculation for u 2 2 : Next notice the pointwise estimate from which we deduce the following two-sided estimate for H − H(0): ) and Continuing the above inequality for u 2 2 we conclude Next we want to get rid of the u ∞ term.For that we note that there exists We use this in the following way, from where we find In total, we have 2 .This is a quadratic inequality in u 2 which implies 2 as claimed.
Step 3.Here we prove There exists . The claim now follows from ≤ C.
Step 4. Next we show in the case After integrating ( 16) over [0, 2π] and taking the real part of the resulting equation we get In order to prove (20) we first suppose d > 0. Then we have on one hand and on the other hand ≥ −|ω|BF Combining the two estimates ( 21), ( 22) and grouping quadratic terms and terms of power 2 . ( The combination of ( 23) and ( 24) leads to Step 5. Finally we prove whenever ζ sign(d) < −C 2 1 d<0 .For this we repeat Step 3 and use in the final estimate that u 2 ≤ D.
Theorem 16 (Schaefer's fixed point theorem).Let X be a Banach space and Φ : X → X be compact.Suppose that the set {x ∈ X : x = λΦ(x) for some λ ∈ (0, 1)} is bounded.Then Φ has a fixed point.
For the next uniqueness result, cf.Theorem 17, let us rewrite the constant D from Theorem 14 as 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 |ζ| 1 is sufficiently large whereas (iii) builds upon f 1 measured in a suitable norm • such that the constant C = C(d, f ) becomes small.This is the case, e.g., if f 2 1 and f 2 remains bounded. and ) are the constants from Theorem 14.
Proof.It suffices to consider the case f = 0.By Theorem 1 we know that (3) has at least one solution u 1 ∈ H 2 per (0, 2π).Now let u 2 ∈ H 2 per (0, 2π) denote an additional solution and define Then u j ∞ ≤ R for j = 1, 2 by Theorem 14, which easily implies Since u j , j = 1, 2 solves the fixed point problem u j = Φ(u j ) we obtain where Next we show 3R 2 L −1 < 1 which implies u 1 = u 2 and thus finishes the proof.To this end we decompose a function v ∈ L 2 (0, 2π) into its Fourier series, i.e., v = ∑ m∈Z v m e ims so that On one hand we get L −1 ≤ 1 since On the other hand, if sign(d) ζ − ω 2 4d > 0, we get In case (i) where sign(d)ζ < ζ * < −C 2 1 d<0 ≤ 0 we use L −1 ≤ 1 and find by the definition of R and ζ * that In case (ii) where sign(d In case (iii) where

PROOF OF THE CONTINUATION RESULTS
In this section we continue to use the notion for the operator L : H 2 per (0, 2π) → L 2 (0, 2π) from Section 4. We also use that L −1 : L 2 (0, 2π) → H 2 per (0, 2π) is bounded and that L −1 : L 2 (0, 2π) → H 1 per (0, 2π) is compact.We first consider continuation from a trivial solution.In order to prove Theorem 6 let us provide the following global continuation theorem.
Theorem 18.Let X be a real Banach space and K ∈ C 1 (R × X, X) be compact.We consider the problem Assume that T(λ 0 , x 0 ) = 0 and that ∂ x T(λ 0 , x 0 ) is invertible.Then there exists a connected and closed set (=continuum) C + ⊂ [λ 0 , ∞) × X of solutions of ( 26) with (λ 0 , x 0 ) ∈ C + .For C + one of the following alternatives holds: If one chooses 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.
There exists also a continuum C − ⊂ (−∞, λ 0 ] × X of solutions of ( 26) with (λ 0 , x 0 ) ∈ C − satisfying one of the alternatives of the theorem.(γ) Alternative (a) of Theorem 18 means that C + is unbounded either in the Banach space direction X or in the parameter direction [λ 0 , ∞) or in both.If unboundedness in the Banach space direction is excluded on compact intervals [λ 0 , Λ], e.g., by a-priori bounds, then unboundedness in the parameter direction follows, i.e., the projection of C + onto [λ 0 , ∞) denoted by pr 1 (C + ) must coincide with [λ 0 , ∞).This is an existence result for all λ ≥ λ 0 which is one aspect of Theorem 6. (δ) Alternative (b) of Theorem 18 means that the continuum C + returns to the λ = λ 0 line at a point Then, as explained before Theorem 16, K is compact and Next we show that ∂ u T(0, u 0 ) is invertible.To this end note that per (0, 2π) and hence, as a compact perturbation of the identity, ∂ 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 4. Now assertion (i) follows from the classical implicit function theorem and Theorem 18 yields that the maximal continuum C + ⊂ [0, ∞) × H 1 per (0, 2π) of solutions ( f 1 , u) of ( 3) with (0, u 0 ) ∈ C + is unbounded or returns to another solution at f 1 = 0.The continuum C + in fact belongs to [0, ∞) × H 2 per (0, 2π) and persists as a connected and closed set in the stronger topology of [0, ∞) × H 2 per (0, 2π).Next we show that the unboundedness of C + coincides with pr 1 (C + ) = [0, ∞).According to Remark 19.(γ) we need to show that unboundedness in the Banach space direction H 1 per (0, 2π) is excluded for f 1 in bounded intervals.To see this suppose that 0 ≤ f 1 ≤ M for all ( f 1 , u) ∈ C + and some constant M > 0.Then, by the a-priori bounds ( 12) and ( 13) from Theorem 14 we get and for all ( f 1 , u) ∈ C + .Hence C + is bounded in the Banach space direction.Assertion (ii) follows in a similar way by using the a-priori bounds of Theorem 14 and the fact that by (3) the bounds for u 2 , u 2 and u ∞ translate into a bound for u 2 .
According to Remark 19.(β) the above line of arguments also yield that the maximal continuum C − ⊂ (−∞, 0] × H 2 per (0, 2π) of solutions of (3) with (0, u 0 ) ∈ C − satisfies pr 1 (C − ) = (−∞, 0] or returns to another solution at f 1 = 0.This finishes the proof. Proof of Corollary 8.The result follows from a combination of Theorem 6 and Theorem 17.For f 1 = 0, i.e. f (s) = f 0 , the abbreviations F, B, C from Theorem 14 and D from Theorem 17 reduce to Hence the constants ζ * , ζ * from Theorem 17 take the form Finally, the conditions (i), (ii), (iii) from the uniqueness result of Theorem 17 translate into the conditions (i), (ii), (iii) from Corollary 8. Now we turn to continuation from a non-trivial solution.Theorem 9 will follow from the Crandall-Rabinowitz Theorem of bifurcation from a simple eigenvalue, which we recall next.
Next we provide the functional analytic setup.Fix the values of d, ω, ζ, f 0 and the function e.If u 0 ∈ H 2 per (0, 2π) is the non-trivial non-degenerate solution of (3) for f 1 = 0 (as assumed in Theorem 9) then for σ ∈ R we denote by u σ (s) := u 0 (s − σ) its shifted copy, which is also a solution of (3) for f 1 = 0. Consider the mapping Then G is twice continuously differentiable.The linearized operator As we shall see there may be more elements in the kernel.Next we fix the value σ 0 (its precise value will be given later) and let H 2 per (0, 2π) = span{u σ 0 } ⊕ Z where, e.g., It will be more convenient to rewrite u = u σ + v with v ∈ Z.In order to justify this, note also that the map (σ, v) → u σ + v defines a diffeomorphism of a neighborhood of (σ 0 , 0) ∈ R × Z onto a neighborhood of u σ 0 ∈ H 2 per (0, 2π) since the derivative at (σ 0 , 0) is given by (λ, ψ) → −λu σ 0 + ψ which is an isomorphism from R × Z onto H 2 per (0, 2π).Now we define which is also twice continuously differentiable and where ∂ ( f 1 ,v) F(σ 0 , 0, 0) is an index-zero Fredholm operator.Our goal will be to solve ( 27) by means of bifurcation theory, where σ ∈ R is the bifurcation parameter.Notice that F(σ, 0, 0) = 0 for all σ ∈ R, i.e., ( f 1 , v) = (0, 0) is a trivial solution of (27).
Proof of Theorem 9.The proof is divided into three steps.
From Theorem 20 we know that σ Therefore, using (33) and find that the condition μ(0) = 0 amounts to assumption (9) of the theorem.
Finally, employing some arguments from spectral theory, we ensure that no other eigenvalue runs into zero.For u = u 1 + iu 2 ∈ H 2 per (0, 2π) let us define the C-linear operator whenever ϕ 1 , ϕ 2 ∈ H 2 per ((0, 2π), R).Since L C u is an index-zero Fredholm operator, its spectrum consists of eigenvalues.The real part of these eigenvalues (weighted with sign(d)) is bounded from below by c ∈ R which is chosen such that Re sign holds.This implies that the resolvent set ρ(L C u ) is non-empty and the compact embedding But from μ(0) = 0 we know that µ(t) = 0 for small |t| > 0 which guarantees that 0 / ∈ σ(L C u(t) ) for 0 < |t| < δ * and δ * sufficiently small.Finally, L u(t) inherits the invertibility of L C u(t) .
both equipped with 2π-periodic boundary conditions.The first equation (37) has a unique solution since the homogeneous equation has a trivial kernel, cf.proof of Theorem 6.Thus v(s) = αe ik 1 s + βe −ik 1 s where α, β ∈ C solve the linear system Solving for α, β leads to the formulae in the statement of the theorem.Since v is the sum of two 2π-periodic complex exponentials and u 0 is a constant we see from (36) that Having determined v we can consider the second equation (38) as an inhomogeneous equation for w.It also has a unique solution since the homogeneous equation is the same as in (37).Since the inhomogeneity is of the form c 1 e i2k 1 s + c 2 e −i2k 1 s + c 3 the solution has the form w(s) = γe i2k 1 s + δe −i2k 1 s + .Moreover, for the determination of

APPENDIX
Here we raise the issue mentioned in Remark 11.(γ) that assumption (10) from Corollary 10 is not satisfied if u 0 is 2π j -periodic and j ∈ N is not a divisor of k 1 .Let us first prove that φ * 0 (spanning ker L * u 0 ) inherits several properties from u 0 (spanning ker L u 0 ).
The proof of (ii) is very similar.Due to the assumption ω = 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.
In a topological sense one can describe lim inf{C + : −1 ∈ N} and lim sup{C + : −1 ∈ N} as in [26].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 (s) the periodic extension of [0, 2π) → C, s → e ik 1 ( )s onto R. Note that

FIGURE 1 . 2 √ 2 4 √ 27 (
FIGURE 1. Curve of squared L 2 -norm of all constant solutions of (3) for f 1 = 0 and f 0 = 1 (green), f 0 = 2 √ 2 4 √ 27 (red) and f 0 = 2 (blue) when ζ ∈ [−1, 5].Turning points (if they exist) are marked with a cross.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 ζ.Starting from f 1 = 0 we use a kind of global implicit function theorem to continue a constant solution u 0 ∈ C of (3) with respect to f 1 .This procedure is analyzed in Theorem 6.The continuation works if the constant solution u 0 ∈ C is non-degenerate in the following sense.

FIGURE 3 .
FIGURE 3. Same situation as in Figure 2. Zoom to the region close to the threshold where the continua change connectivity.

Figure 4
Figure 4 illustrates the same application, but depicted from a different angle and with more values of ζ.Repeating the simulation with d = 0.1 (anomalous dispersion) instead of d = −0.1 (normal dispersion) did not change the picture essentially.Figures2, 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 13 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 → u( f 1 ).The type of local extremum is described by the sign of the second derivative d2

FIGURE 4 .
FIGURE 4. Same situation as in Figure 2, but depicted from a different angle and with more values of ζ.

FIGURE 5 .
FIGURE 5. Sign of the second derivative of f 1 → u( f 1 ) 2 2 at f 1 = 0; blue=positive, red=negative.A second sign-change (in fact a singularity changing from −∞ to +∞) occurs at the first turning point.Then, the next sign-change occurs on the part of the branch between the two turning points at ζ ≈ 3.34.Finally, the second turning point generates the last sign-change from −∞ to +∞.Clearly, the changes in the nature of the local extremum of f 1 → u( f 1 ) 2 2 at f 1 = 0 do not correspond to the topology changes of the solution continua which occur near the threshold value ζ * ∈ (3.1344, 3.1359).Next, we keep the parameters d = −0.1,f0 = 2, k 1 = 1 but choose ω = 0 instead of ω = 1.Recall that for ω = 0 there is a plethora of non-trivial solutions of (2) for f 1 = 0, cf.[6],[13].In fact, this time we find additional primary and secondary bifurcation branches for f 1 = 0 which are illustrated in Figure6in grey and brown, respectively.Bifurcation points are shown as grey dots.The bifurcation branches consist of non-trivial solutions.Further, some numerical approximations of the two-sided maximal continua C obtained by continuation of trivial or non-trivial solutions for different values of the detuning ζ are shown.If we start from a constant solution at f 1 = 0, then C ± are described by Theorem 6.Likewise, if we start from a non-constant solution at f 1 = 0 which has no in grey and brown, respectively.Bifurcation points are shown as grey dots.The bifurcation branches consist of non-trivial solutions.Further, some numerical approximations of the two-sided maximal continua C obtained by continuation of trivial or non-trivial solutions for different values of the detuning ζ are shown.If we start from a constant solution at f 1 = 0, then C ± are described by Theorem 6.Likewise, if we start from a non-constant solution at f 1 = 0 which has no smaller period than 2π, then C ± are described by Theorem 9.In both cases, C ⊃ C + ∪ C − by Proposition 12, 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 ζ where different phenomena occur.

Proof of Theorem 13 .
Let us fix all parameters d, ω, ζ, k 1 and f 0 and consider u :f 1 → u( f 1 ) as a function mapping the parameter f 1 ∈ [− 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 6. Similarly we consider the functions v : f 1 → du( f 1 ) d f 1 and w : f 1 → d 2 u( f 1 ) ) + |v| 2 ds and the differential equations for v, w at f 1 = 0 are given by of γ, δ are irrelevant and only the value of matters.Using|v| 2 = |α| 2 + |β| 2 + 2 Re(αβe i2k 1 s ), v 2 = α 2 e i2k 1 s + β 2 e −i2k 1 s + 2αβwe find from (38) that the equation determining is(ζ − i) − 4u 0 (|α| 2 + |β| 2 ) − 4u 0 αβ − 2|u 0 | 2 − u 2 0 = 0.Since this is an equation of the form x + y = z with x, y, z given in the statement of the theorem we find the solution formula = −zy+zx |x| 2 −|y| 2 .Finally, only the constant contributions from w and |v| 2 contribute to the integral in the formula (36) ford 2 d f 2 1 u( f 1 )22 and lead to the claimed statement of the theorem.