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Rippling rectangular waves for a modified Benney equation

  • Tomoyuki MiyajiEmail author
  • Toshiyuki Ogawa
  • Ayuki Sekisaka
Open Access
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Abstract

One parameter family of rectangular periodic traveling wave solutions are known to exists in a perturbed system of the modified KdV equation. The rectangular periodic traveling wave consists basically of front and back transitions. It turns out that the rectangular traveling wave becomes unstable as its period becomes large. More precisely, torus bifurcation occurs successively along the branch of the rectangular traveling wave solutions. And, as a result, a “rippling rectangular wave” appears. It is roughly the rectangular traveling wave on which small pulse wave trains are superimposed. The bifurcation branch is constructed by a numerical torus continuation method. The instability is explained by using the accumulation of eigenvalues on the essential spectrum around the stationary solutions. Moreover, the critical eigenfunctions which correspond to the torus bifurcation can be characterized theoretically.

Keywords

Modified KdV equation Benney equation Rectangular traveling wave Torus bifurcation. 

Mathematics Subject Classification

34A26 37L15 

1 Introduction

We study a perturbed system:
$$\begin{aligned} \partial _t u - \frac{1}{3}\partial _x u^3 + \partial _x^3 u + \varepsilon \left( \partial _x^2 u + \partial _x^4 u - \frac{\alpha }{3}\partial _x^2 u^3 \right) = 0, \end{aligned}$$
(1)
of the modified KdV (m-KdV) equation:
$$\begin{aligned} \partial _t u - \frac{1}{3}\partial _x u^3 + \partial _x^3 u = 0, \end{aligned}$$
(2)
on the whole line. Here, \(u = u(t, x)\) is a real-valued function of time t and space \(x \in \mathbb {R}\). Moreover, the constant \(\varepsilon \) is supposed to be positive. The m-KdV equations are well-known integrable systems appearing in the study of the KdV equation. There are two different types depending on the coefficient of the nonlinear term. In fact, the m-KdV equation (2) has traveling front solutions \(u(z)=\pm \sqrt{3c}\tanh \left( \sqrt{c/2}(x+ct) \right) \) for an arbitrary \(c>0\). They are sometimes called “kink” and “anti-kink”. On the other hand, the different version of the m-KdV equation, which is obtained by replacing the term \(-(1/3)\partial _x u^3\) in (2) by \((1/3)\partial _x u^3\), has solitary wave solutions.

The m-KdV equation is also obtained from a long wave approximation of a certain fluid dynamics problem. By taking higher order approximation terms we can also obtain (1) instead of (2). In fact, Komatsu and Sasa [12] obtained both (2) and (1) from the so-called optimal velocity model which characterizes traffic congestion. The unknown variable u means the density of the traffic vehicles in their formulation. Therefore, sequences of kink and anti-kink can be considered as the bands of traffic jams.

Now, the behavior of the solutions of the equations (2) and (1) are similar in the sense that both have traveling front solutions and by taking \(\varepsilon \) smaller the front solutions to (1) become close to that of (2) as we see later. However, the perturbed equation has a remarkable property in the following sense. In fact, only one traveling front solution can persist for \(\varepsilon >0\) among the uncountably many front solutions to (2). We can say that the velocity and amplitude of the front solution to (2) are chosen by the balance of dissipative perturbation terms.

The similar kind of observation and analysis can be found in [1, 11, 15]. They deal with the so-called Benney equation:
$$\begin{aligned} \partial _t u + \frac{1}{2}\partial _x u^2 + \partial _x^3 u + \varepsilon \left( \partial _x^2 u + \partial _x^4 u \right) = 0, \end{aligned}$$
(3)
which is obtained by adding dissipative perturbations to the KdV equation:
$$\begin{aligned} \partial _t u + \frac{1}{2}\partial _x u^2 + \partial _x^3 u = 0 . \end{aligned}$$
(4)
Let us roughly explain the reason why the amplitude of the KdV soliton solution:
$$\begin{aligned} u(t,x)=A~\mathrm {sech}^2\left( \sqrt{2A}(x+A/3 \ t) \right) , \end{aligned}$$
(5)
where A is an arbitrary positive number, is selected to survive for the perturbations. Consider the time development of the two conservations \(M=\int u dx\) and \(W=\int u^2 dx\) for the KdV equation. We can obtain the following from (3):
$$\begin{aligned} \displaystyle \frac{dM}{dt}= & {} 0 \ , \end{aligned}$$
(6)
$$\begin{aligned} \displaystyle \frac{dW}{dt}= & {} \varepsilon \left( \int (\partial _x u)^2 dx - \int (\partial _x ^2 u)^2 dx \right) \ . \end{aligned}$$
(7)
It is clear that both M and W are constants along the traveling wave solution, therefore we can determine the amplitude by plugging (5) to \(dW/dt=0\). We are going to discuss this amplitude selection later mathematically for (1) and even for the periodic traveling wave solutions. In fact, we can conclude there is a one-parameter family of periodic traveling wave solutions for each small \(\varepsilon \) parameterized by the wavelength.

However, most significant point in this paper is not this amplitude selection of periodic solutions for each period but the stability change of the periodic traveling wave solutions. We can observe a new solution bifurcates by the torus bifurcation along the branch of the periodic traveling wave solutions. It is roughly the rectangular periodic traveling wave solution on which small pulse wave trains are superimposed. And, moreover, these superimposed pulses move at the velocity which is different from the background rectangular wave. Therefore, it is not a traveling wave after the bifurcation. We call this solution the rippling rectangular wave. See Fig. 1.

One might consider that it is reasonable for the equation (1) to have such type of rippling rectangular wave by the following heuristic argument. Let us linearize (1) at the lowest and highest flat levels \(\pm ~u_0\) since the periodic traveling wave becomes close to the kink and anti-kink as its spatial period becomes large. By plugging \(u=\tilde{u}\pm u_0\) into the equation and neglecting the higher order terms we obtain
$$\begin{aligned} \partial _t \tilde{u} \pm u_0^2 \partial _x \tilde{u} + \partial _x^3 \tilde{u} + \varepsilon \left( \partial _x^2 \tilde{u} + \partial _x^4 \tilde{u} - 3\alpha u_0^2 \partial _x^2 \tilde{u}\right) =0. \end{aligned}$$
(8)
Therefore, we may observe a similar instability about \(\tilde{u}=0\) as the Benney equation so that pulse wave trains may bifurcate depending on the length of the flat part, \(\alpha \), and \(u_0\) as well.
Fig. 1

Bird’s eye view of solutions of (1) with \(\alpha = 0, \varepsilon = 5 \times 10^{-3}\). a A periodic traveling wave for \(L=30\). b A rippling rectangular wave for \(L=86\)

To approach the bifurcation we study solutions that are L-periodic with respect to x. In other words, we study (1) on the interval (0, L) under the periodic boundary condition. We show that the rectangular waves are unstable for sufficiently large L provided that \(\alpha \) is less than some threshold. Moreover, we prove that the distance between two successive torus bifurcation points tends to \(2 \pi / \sqrt{1 - \alpha u_0^2}\) as L goes to the infinity. The proofs of these theorems are based on a topological argument, which is simpler than an analytic approach and easy to understand the connection with numerical simulations. In addition to the theoretical results, we also show numerical results for (1) with \(\alpha = 0\). We compute the torus bifurcation points, from which the rippling rectangular waves bifurcate, as well as critical eigenfunctions. We compute numerically the branches of the rippling rectangular waves by applying the so-called parametrization method and the predictor-corrector method. We also compute the torus bifurcation curves drawn on the \((L, \alpha )\)-plane to observe the dependency on the parameter \(\alpha \).

It may be noticed that the parameter \(\alpha = 2/3\) in the case of the reduced equation from the optimal velocity model by Komatsu and Sasa [12]. Therefore, the rippling waves may not be observed in their case. That means the nonlinear diffusion term \(-~\alpha \partial _x^2 u^3\) plays a role of stabilizing the rectangular wave. We have studied (1) with \(\alpha < 2/3\) from the mathematical interest. We, however, believe that whole information of the bifurcation structure may be useful to really understand the model.

The rest of this paper is organized as follows. We present our main theorems in the next section. We postpone the proofs of the theorems to Sect. 4 for the sake of readability. Section 3 is devoted to numerical results. The results for Sects. 24 are discussed in Sect. 5. The existence of the rectangular waves is proved in Sect.  “Appendix A”. Finally, we present our numerical methods in Sect.  “Appendix B”.

2 Traveling waves

In this section, we consider the existence and stability of traveling wave solutions for (1). Introducing a moving frame by \(\xi = x + c t\), we can rewrite (1) as
$$\begin{aligned} \partial _{t}u + c \partial _{\xi } u + \partial ^{3}_{\xi }u -\frac{1}{3}\partial _{\xi }u^{3} + \varepsilon \left( \partial ^{2}_{\xi }u + \partial ^{4}_{\xi }u -\frac{\alpha }{3}\partial ^{2}_{\xi }u^{3}\right) =0. \end{aligned}$$
(9)
Traveling waves then are steady states of (9), that is, they satisfy the ordinary differential equation(ODE):
$$\begin{aligned} -u_{\xi \xi \xi } + u^{2}u_{\xi } - cu_{\xi } - \varepsilon \left( u_{\xi \xi } + u_{\xi \xi \xi \xi } - 2\alpha u u^{2}_{\xi } -\alpha u^{2}u_{\xi \xi }\right) = 0. \end{aligned}$$
(10)
(9) can be rewritten as the first-order system in \({\mathbb {R}}^{4}\):
$$\begin{aligned} \left\{ \begin{aligned} u'&= v,\\ v'&= w,\\ w'&= z,\\ z'&= \frac{1}{\varepsilon }\left( (u^{2}-c)v - z \right) - w + \alpha (2uv^{2} + u^{2} w), \end{aligned} \right. \end{aligned}$$
(11)
or simply
$$\begin{aligned} U'=F(U,c,\varepsilon ,\alpha ), \end{aligned}$$
(12)
where \('=\frac{d}{d\xi }\) and \(U=(u,v,w,z) \in {\mathbb {R}}^{4}\).

The following theorem shows that (11) has a family of periodic orbits and two heteroclinic orbits.

Theorem 1

Suppose \(\alpha \ge 0\). There are small constant \(\varepsilon _{*}\) and smooth functions \(c(\varepsilon ,L)\) and \(c_{\infty }(\varepsilon ), 0 \le \varepsilon < \varepsilon _{*}\), such that the followings are satisfied for all \(0 \le \varepsilon < \varepsilon _{*}\);
  1. 1.

    There exists \(L_0 \ge 0\) such that for any \(L \in (L_0, \infty )\), (11) with \(c = c(\varepsilon , L)\) has a periodic solution \(h_{per}(\xi ; c(\varepsilon ,L),L)\).

     
  2. 2.
    If \(c=c_{\infty }(\varepsilon )\), then (11) has two heteroclinic solutions \(h_{f}(\xi )\) and \(h_{b}(\xi )\) satisfying
    $$\begin{aligned}&\lim _{\xi \rightarrow -\infty }h_{f}(\xi )=-U_{0}, \ \lim _{\xi \rightarrow +\infty }h_{f}(\xi )=U_{0},\\&\lim _{\xi \rightarrow -\infty }h_{b}(\xi )=U_{0}, \ \lim _{\xi \rightarrow +\infty }h_{b}(\xi )=-U_{0}, \end{aligned}$$
    where \(\pm U_{0}(c_{\infty }(\varepsilon )) = (\pm u_{0}, 0, 0, 0)\) are equilibria for (11) with \(u_{0}=\sqrt{3c_{\infty }(\varepsilon )}\).
     
  3. 3.
    \(c(\varepsilon ,L)\) converges to \(c_{\infty }(\varepsilon )\) as \(L \rightarrow \infty \), and hence,
    $$\begin{aligned} \lim _{L \rightarrow \infty } d_{H}(\{h_{per}(\xi ;c(\varepsilon ,L),L) \} , \varGamma )=0, \end{aligned}$$
    (13)
    where \(\varGamma = \mathop {\mathrm {cl}}\nolimits (\{ h_{f}(\xi ) \mid \xi \in \mathbb {R}\} \cup \{ h_{b}(\xi ) \mid \xi \in \mathbb {R}\} )\) and \(d_{H}\) is the Hausdorff metric.
     

We present a proof of this theorem, which is based on [15] in Appendix.

Theorem 1 implies that there exists a family of stationary solutions \(\{ u_{per}(\xi ,c; L) \}_{L \in \Lambda }\) for (9) with the periodic boundary condition on \(I_{L} = (0, L)\). Indeed, \(u_{per}\) corresponds to the first component of \(h_{per}\). Similarly, there are two front solutions \(u_{f}\) and \(u_{b}\) for (9) with the velocity \(c_{\infty }(\varepsilon )\). They correspond to the first components of \(h_{f}\) and \(h_{b}\), respectively.

In the following arguments, we fix \(\varepsilon \in (0, \varepsilon _{*})\), and we put \(c_{*}=c_{\infty }(\varepsilon )\).

Next, we consider the linearized stability problem of the stationary solutions \(\{u_{per}(\xi ,c(L);L \} \) for (9). The linearized equation of (9) associated with \(u_{per}(\xi ,c(L);L)\) is given by
$$\begin{aligned} p_{t}= & {} -p_{\xi \xi \xi }+ (u^{2} -c(L))p_{\xi } + 2uu_{\xi }p \nonumber \\&-\varepsilon \left( p_{\xi \xi } + p_{\xi \xi \xi \xi }-\alpha (u^{2} p)_{\xi \xi } \right) , \end{aligned}$$
(14)
where \(u = u_{per}(\xi , c(L); L)\). We denote the linear operator in the right hand side by \({\mathcal {L}}_{per, L}\), that is,
$$\begin{aligned} {\mathcal {L}}_{per,L}\ p= & {} -p_{\xi \xi \xi } + (u^{2} -c(L))p_{\xi } + 2uu_{\xi }p \\&- \varepsilon \left( p_{\xi \xi } + p_{\xi \xi \xi \xi } -2\alpha u_{\xi }^{2}p - \alpha u^{2} p_{\xi \xi \xi } \right) . \end{aligned}$$
\({\mathcal {L}}_{per, L}\) is an operator on \(L^{2}(I_{L})\) whose domain is the Sobolev space
$$\begin{aligned} H^4(I_{L}) \cap \left\{ u \mid u^{(i)}(0)= u^{(i)}(L), i=0,1,2,3 \right\} , \end{aligned}$$
where \(I_{L} =[0,L]\). We consider the eigenvalue problem of \({\mathcal {L}}_{per, L}\):
$$\begin{aligned} {\mathcal {L}}_{per, L}\ p = \lambda p. \end{aligned}$$
We deonte the spectrum of \({\mathcal {L}}_{per, L}\) by \(\sigma ({\mathcal {L}}_{per, L})\). In the above setting, \(\sigma ({\mathcal {L}}_{per, L})\) consists of only discrete eigenvalues because \(I_{L}\) is compact. One of our results induces the spectral instability of periodic solutions \(u_{per}\) with long period, that is, at least one eigenvalue of \({\mathcal {L}}_{per}\) is contained in the right-half plane.

Theorem 2

There exists \(L_{*}>0\) such that \(u_{per}(\xi ,c(L);L)\) are unstable for any \(L > L_{*}\) and sufficiently small \(\alpha \ge 0\).

Theorem 3

Suppose \(\alpha \) is sufficiently small and nonnegative. There exists \(L_{\infty } > 0\) and a sequence \(\{ h_{n} \}_{n=1}^{\infty }\) such that the intersection of \(\sigma ({\mathcal {L}}_{per, L_n})\) and \(\mathrm {i}\mathbb {R}\setminus \{ 0 \}\) is nonempty for any \(n = 1, 2, \dots \), where \(L_n\) is given by
$$\begin{aligned} L_{n} = L_{\infty } + \dfrac{2 \pi n}{\sqrt{1 - \alpha u_0^2}} + h_{n}, \ n=1,2,\cdots , \end{aligned}$$
(15)
and \(h_{n}\) converges monotonically to zero as n tends to infinity, where \(u_0 = u_0(c_{*})\).

This theorem suggests that \(u_{per}\) may undergo the Hopf bifurcation at \(L = L_n\) for each n, and \(L_{n+1} - L_n \approx 2 \pi / \sqrt{1 - \alpha u_0^2}\) holds for sufficiently large n. Here, the Hopf bifurcation in the moving coordinates corresponds to the torus bifurcation in the original coordinates.

Remark 1

It should be noticed that \(\alpha \) needs not to be nonnegative. In fact, we can prove Theorem 1 even for negative \(\alpha \) under the condition that some quantities are positive. See the denominators in (37) and (39) in Appendix A in detail. We have, however, assumed that \(\alpha \) is nonnegative for simplicity in Theorems 13.

We shall show Theorems 2 and 3 in the later section. Instead, we introduce the numerical results in the following section to see the meanings of these theorems.

3 Numerical results

We present numerical results for (1) with the periodic boundary condition on (0, L). Especially, we focus on the cases where \(L < 100\). In this section, we set parameters
$$\begin{aligned} (\alpha , \varepsilon ) = (0, 5 \times 10^{-3}), \end{aligned}$$
unless otherwise mentioned. We regard a rectangular wave with the velocity c for (1) as a time-periodic solution with the period L / |c| because this view is useful for our numerical bifurcation analysis. The rest of this section is organized in the following way. First, we present a bifurcation diagram of the rectangular wave for (1). In the bifurcation diagram below, we find torus bifurcation points, which are nothing but the Hopf bifurcation points in the moving coordinates (9). Next, we focus on individual solutions appearing from the torus bifurcation points. Finally, we present a result of two-parameter continuation for the torus bifurcation points with respect to \((L, \alpha )\), keeping \(\varepsilon = 5 \times 10^{-3}\) fixed. Details of numerical methods are presented in the appendix of this paper.
Figure 2 presents a bifurcation diagram. The horizontal and vertical axes indicate the space size L and \(L^2\)-norm of the imaginary part of the first Fourier mode of the solution [\(L^2\)-norm of \(v_1(\theta )\) in (44) below], respectively . The black line indicates the branch of rectangular waves, and the red or blue lines indicate the branch of bifurcating solutions. The rectangular wave loses its stability at \(L \approx 67.07322\) by the torus bifurcation. Moreover, there are eight torus bifurcation points, which are labeled “TR” in the diagram, on the branch of rectangular wave in \(65< L < 100\). Table 1 presents the values of L at the torus bifurcation points. The solutions on the branch coming from TRn are also referred to as TRn \((n = 1, 2, \dots 8)\).
Fig. 2

A bifurcation diagram of a traveling solution and rippling rectangular waves of (1) with \(\alpha = 0, \varepsilon = 5 \times 10^{-3}\). The horizontal axis is L and the vertical axis is the \(L^2\)-norm of \(v_1(\theta )\)

Table 1

Torus bifurcation points of the rectangular wave computed by using AUTO-07P [4] with \(N = 100\) and 200, where N is the truncation wave number

Label

L

# of maxima

Type

TR1

\(\underline{67.07322}0970\)

7

In-phase

TR2

\(\underline{69.500}543921\)

6

Anti-phase

TR3

\(\underline{70.252}425300\)

8

Anti-phase

TR4

\(\underline{74.827}328777\)

9

In-phase

TR5

\(\underline{80.010}856668\)

10

Anti-phase

TR6

\(\underline{85.5}20323492\)

11

In-phase

TR7

\(\underline{91.22}8427677\)

12

Anti-phase

TR8

\(\underline{97.0}71317477\)

13

In-phase

The displayed digits are obtained for \(N = 200\), and the results of \(N = 100\) and 200 agree within the underlined digits

Figure 3 presents profiles of the critical eigenfunctions at the torus bifurcations. The number of maxima (and minima) increases as L increases, except first two bifurcations.
Fig. 3

Profiles of critical eigenfunctions at TR bifurcation points of the rectangular wave for \(\alpha = 0, \varepsilon = 5 \times 10^{-3}\). a TR1: \(L \approx 67.07322\). b TR2: \(L \approx 69.500\). c TR3: \(L \approx 70.252\). d TR4: \(L \approx 74.827\). e TR5: \(L \approx 80.010\). f TR6: \(L \approx 85.5\). g TR7: \(L \approx 91.22\). h TR8: \(L \approx 97.0\)

We classify the bifurcating solutions, which we call rippling rectangular waves, into two types: in-phase and anti-phase solutions, respectively. Let us illustrate it by Figs. 4 and 5. First, Fig. 4 illustrates snapshots of an in-phase solution for \(L = 86\):
  1. 1.

    At \(t = 1.8\), there are two humps on the lower plateau and two hollows on the higher plateau.

     
  2. 2.

    At \(t = 2.7\), an additional hump and hollow appear simultaneously from the interface between lower and higher plateaus.

     
  3. 3.

    At \(t = 3.6\), the leftmost hump and hollow reach the cliff and disappear simultaneously.

     
  4. 4.

    At \(t = 4.5\), the solution recovers a similar profile as \(t = 1.8\), while it is shifted to the left.

     
The solution repeats these process. We call this type of solutions in-phase because the humps and hollows are synchronous.
Next, Fig. 5 illustrates snapshots of an anti-phase solution for \(L = 87.675611\):
  1. 1.

    At \(t = 1.6\), there are two humps and two hollows on the lower and higher plateaus, respectively.

     
  2. 2.

    At \(t = 2.4\), an additional hump appears on the lower plateau.

     
  3. 3.

    At \(t = 3.2\), the leftmost hump disappears at the right cliff. Again, there are two humps and two hollows on the lower and higher plateaus, respectively.

     
  4. 4.

    At \(t = 4.0\), an additional hollow appears on the higher plateau.

     
The solution repeats these process. We call this type of solutions anti-phase because the humps and hollows appear and disappear alternately.
Figure 6 presents profiles of the rippling rectangular waves at \(L = 100\). Each of them has different number of humps. Figure 6a–d present the rippling rectangular waves TR1, TR4, TR6, and TR8, which are in-phase solutions, while (e–h), which appears from TR2, TR3, TR5, and TR7, are anti-phase ones.
Fig. 4

Snapshots of a stable rippling rectangular wave for \(L = 86, \alpha = 0, \varepsilon = 5.0 \times 10^{-3}\). An in-phase solution is presented. a \(t = 1.8\). b \(t = 2.7\). c \(t = 3.6\). d \(t = 4.5\)

Fig. 5

Snapshots of a stable rippling rectangular wave for \(L = 87.675611, \alpha = 0, \varepsilon = 5.0 \times 10^{-3}\). An anti-phase solution is presented. a \(t = 1.6\). b \(t = 2.4\). c \(t = 3.2\). d \(t = 4.0\)

Fig. 6

Profiles of rippling rectangular waves for \(L = 100, \alpha = 0\), and \(\varepsilon = 5 \times 10^{-3}\). The label TRn means that it bifurcates by the torus bifurcation labeled TRn (\(n = 1, 2, \dots 8\)). a branch TR1. b TR4. c TR6. d TR8. e TR2. f TR3. g TR5. h TR7

Next, we consider the case where \(\alpha > 0\). Figure 7 presents profiles of traveling waves at \(L=65\) for some values of \(\alpha \). As \(\alpha \) increases, the amplitude of the traveling wave decreases. Figure 8 presents a two-parameter bifurcation diagram, in which the torus bifurcation points in Fig. 2 are continued in the \((L, \alpha )\)-plane. The value of L at each bifurcation point is almost monotone as a function of \(\alpha \) in the parameter length under consideration. An exception is the curve of TR2: it has a fold point and becomes multivalued (See Fig. 8b). As shown in the previous section, no torus bifurcation occurs if \(\alpha \) is sufficiently large. It can be considered that the existence of the fold point reflects this fact. Similarly, other torus bifurcation curves may possess a fold point, although we have not checked it.
Fig. 7

Profiles of traveling waves for \(L = 65\) and \(\alpha = 0, 1/3, 2/3\), and 1

Fig. 8

Two-parameter bifurcation diagram of the torus bifurcation drawn on the \((L, \alpha )\)-plane. \(\varepsilon = 5 \times 10^{-3}\). a Bifurcation diagram. b A blowup view

4 Proof of theorem

From Theorem 1, there are two front solutions \(u_{f}(x+c_{\infty }(\varepsilon )t)\) and \(u_{b}(x+c_{\infty }(\varepsilon )t)\) connecting \(\pm u_{0}(c_{\infty }(\varepsilon ))\), and \(u_{0}=u_{0}(c_{*}), \ c_{*}=c(\varepsilon )\) is unique with respect to fixed \(\varepsilon \), that is, \(u_{0}=\sqrt{3c_{\infty }(\varepsilon )}\).

Let \({\mathcal {L}}_{0,L}\) be the linearized operator about \(u_{0}\) given by
$$\begin{aligned} {\mathcal {L}}_{0,L}\ p=-p_{\xi \xi \xi } + (u_{0}^{2} -c_{*})p_{\xi } - \varepsilon (p_{\xi \xi } + p_{\xi \xi \xi \xi }- \alpha u_{0}^{2} p_{\xi \xi }), \end{aligned}$$
where \(\xi \in (0, L)\). Remark that the linearized operator about \(-u_0\) coincides with \({\mathcal {L}}_{0, L}\).

Theorem 2 can be proved by the following two key facts. First, the uniform stationary solutions \(\pm u_{0}\) are unstable on the whole line \({\mathbb {R}}\) if the parameter \(\alpha >0\) is sufficiently small since the essential spectrum intersects with the right half plane. Second, eigenvalues of the linearized operator \({\mathcal {L}}_{per,L}\) accumulate to the essential spectrum of \({\mathcal {L}}_{0,L}\) when the period L of \(u_{per}\) tends to infinity. This is because the periodic orbit corresponding to the traveling wave solution stays close to the uniform stationary solution for a longer and longer period as we take the period L larger and larger. These two are similar to the discussion on the stability analysis for the periodic orbit close to a homoclinic orbit by Gardner [6]. Sandstede and Scheel [16] also discussed about the similar problem with an extension. In this paper, we consider a family of periodic orbits close to a heteroclinic cycle; This is different from the previous studies. Moreover, we take a topological approach, which has been developed in [19] and [18]. This simplifies the proof of Theorems 2 and 3, and this allows us to understand the connection of theoretical and numerical results as we shall discuss in Sect. 5.

First, we consider the eigenvalue problem of \({\mathcal {L}}_{0,L}\),
$$\begin{aligned} {\mathcal {L}}_{0,L}\ p=\lambda p. \end{aligned}$$
In a similar way to the derivation of (11), we rewrite this equation in the normal form:
$$\begin{aligned} Y' = A_{0}(\lambda )Y, \ Y \in {\mathbb {C}}^{4}, \end{aligned}$$
(16)
where
$$\begin{aligned} A_{0}(\lambda )= \begin{pmatrix} 0 &{} 1 &{} 0 &{} 0\\ 0 &{} 0 &{} 1 &{} 0\\ 0 &{} 0 &{} 0 &{} 1\\ -\frac{\lambda }{\varepsilon } &{} \frac{(u_{0}^{2}-c_{*})}{\varepsilon } &{} \alpha u_{0}^{2} -1 &{} -\frac{1}{\varepsilon } \end{pmatrix}. \end{aligned}$$
(17)
The characteristic polynomial of \(A_{0}(\lambda )\) is given by
$$\begin{aligned} K_{\alpha }(\mu )=\mu ^{4} +\frac{1}{\varepsilon }\mu ^{3} +(1-\alpha u_{0}^{2})\mu ^{2} +\left( \frac{c_{*}-u_{0}^{2} }{\varepsilon } \right) \mu + \frac{\lambda }{\varepsilon }. \end{aligned}$$

Definition 1

The essential spectrum of \({\mathcal {L}}_{0,L}\) is defined by
$$\begin{aligned} \varSigma ^{0}_{ess}:=\left\{ \lambda \mid \mathop {\mathrm {spec}}\nolimits A_{0}(\lambda ) \cap i{\mathbb {R}} \ne \emptyset \right\} . \end{aligned}$$
(18)

Remark that \(\varSigma _{ess}^{0}\) coincides with the essential spectrum of \({\mathcal {L}}_{0,L}\) even if the operator \({\mathcal {L}}_{0,L}\) is defined on \(L^{2}({\mathbb {R}})\). Therefore, the following lemma holds.

Lemma 2

The essential spectrum \(\varSigma ^{0}_{ess}\) of \({\mathcal {L}}_{0,L}\) is given by
$$\begin{aligned} \varSigma ^{0}_{ess}= \left\{ \lambda \mid \lambda = \varepsilon ((1-\alpha u_{0}^{2}) k^{2} - k^{4}) + (k^{2} + u_{0}^{2}-c_{*} )ik, \ k \in {\mathbb {R}} \right\} . \end{aligned}$$
Hence, if \(\alpha u_0^2 < 1\), then
$$\begin{aligned} \varSigma ^{0}_{ess} \cap \left\{ \lambda \in {\mathbb {C}} \mid \mathop {\mathrm {Re}}\nolimits \lambda > 0 \right\} \ne \emptyset . \end{aligned}$$

Proof

It is immediate from the dispersion relation of \({\mathcal {L}}_{0, L}\). \(\square \)

We remark that \(\alpha \) is the control parameter, and hence, Lemma 2 means that constant steady states \(\pm u_{0}\) are unstable for sufficiently small \(\alpha \). Let \(\mu _{j}(\lambda )\) be eigenvalues of \(A_{0}(\lambda )\) ordered as
$$\begin{aligned} \mathrm {Re}\,\mu _{j}(\lambda ) > \mathrm {Re}\,\mu _{j+1}(\lambda ), \end{aligned}$$
for \(j = 1, 2, 3\). Then the following holds.

Lemma 3

Let \(\lambda = \lambda (k) \in \varSigma _{ess}^{0}\),
$$\begin{aligned} \lambda (k)= \varepsilon ((1-\alpha u_{0}^{2}) k^{2} - k^{4}) + (k^{2} + u_{0}^{2}-c_{*} )ik, \end{aligned}$$
for an arbitrary \(k \in \mathbb {R}\). There is just one eigenvalue \(\mu ^{*}(\lambda ) = \mathrm {i}\omega (\lambda )\) of \(A_{0}(\lambda )\) satisfying \(\mathrm {Re}\,\mu ^*(\lambda )=0\) and \(\omega (\lambda ) = \mathop {\mathrm {Im}}\nolimits \mu ^{*}(\lambda ) =k\). Moreover, if \(\alpha u_0^2 < 1\), then
$$\begin{aligned} \left. \dfrac{d \omega }{d \lambda }(\lambda )\right| _{\lambda = \lambda _{*} } \notin \mathrm {i}\mathbb {R}, \end{aligned}$$
for any \( \lambda _{*} \in S_{per}:=\varSigma _{ess}^{0} \setminus \{ 0, \lambda (\pm \sqrt{(1-\alpha u_0^2)/2}) \}\).

Proof

By a simple calculation, we obtain
$$\begin{aligned} \dfrac{d \omega }{d \lambda }(\lambda (k))=\dfrac{1}{2k\varepsilon (1-\alpha u_{0}- 2k^{2}) + i(3k^{2}+u_{0}^{2} -c_{*} )}, \end{aligned}$$
Therefore, the assertion holds. \(\square \)
Let us also rewrite the eigenvalue problem of \({\mathcal {L}}_{per,L}\)
$$\begin{aligned} {\mathcal {L}}_{per,L}\ p = \lambda p, \end{aligned}$$
as
$$\begin{aligned} Y' = A_{per}(\xi ;\lambda ,L)Y, \ Y \in {\mathbb {C}}^{4}. \end{aligned}$$
(19)
Here,
$$\begin{aligned} A_{per}(\xi ;\lambda ,L)= \begin{pmatrix} 0 &{} 1 &{} 0 &{} 0\\ 0 &{} 0 &{} 1 &{} 0\\ 0 &{} 0 &{} 0 &{} 1\\ -\frac{\lambda }{\varepsilon } &{} \frac{(u^{2}-c(L))}{\varepsilon } &{} \alpha u^{2} -1 &{} -\frac{1}{\varepsilon } \end{pmatrix}, \end{aligned}$$
(20)
and \(u = u_{per}(\xi , c(L); L)\). We also write the eigenvalue problem for the front traveling wave solution \(u_{*}(\xi ), \ *=f,b\) as follows:
$$\begin{aligned} Y' = A_{*}(\xi ;\lambda )Y, \ Y \in {\mathbb {C}}^{4}, *=f,b. \end{aligned}$$
(21)
It clearly holds that
$$\begin{aligned} \lim _{\xi \rightarrow \pm \infty }A_{*}(\xi ;\lambda )=A_{0}(\lambda ). \end{aligned}$$
(22)
Let us define the subsets of \({\mathbb {R}}^{4}\) as
$$\begin{aligned}&\varPi _{f}^{+}= \left\{ (u,v,w,z) \mid v=\gamma , u_{0}-\gamma< u< u_{0}+\gamma , |w|< \gamma , |z|< \gamma \right\} ,\\&\varPi _{f}^{-}= \left\{ (u,v,w,z) \mid v=\gamma , -u_{0} + \gamma< u< -u_{0}+\gamma , |w|< \gamma , |z| < \gamma \right\} , \end{aligned}$$
and
$$\begin{aligned}&\varPi _{b}^{+}= \left\{ (u,v,w,z) \mid v=-\gamma , u_{0}-\gamma< u<u_{0}+\gamma , |w|< \gamma , |z|< \gamma \right\} ,\\&\varPi _{b}^{-}= \left\{ (u,v,w,z) \mid v=-\gamma , -u_{0} - \gamma< u< -u_{0}+\gamma , |w|< \gamma , |z| < \gamma \right\} . \end{aligned}$$
We take \(\gamma >0\) sufficiently small so that the sets \(\varPi _{f}^{\pm }\) and \(\varPi _{b}^{\pm }\) are local cross sections for the heteroclinic orbits \(h_{f}\) and \(h_{b}\). Then it is clear that \(\varPi _{f}^{\pm }\) and \(\varPi _{b}^{\pm }\) are also local cross sections for \(h_{per}(\xi ;c(L),L)\) as well for sufficiently large L.

Now let \(\xi _{f,b}^{\pm }(L)\) are the times when the periodic orbit crosses the local sections: \(h_{per}(\xi _{i}^{j}(L);c(L),L) \in \varPi _{i}^{j}, i=f,b, \ j=+,-\) with \(0\le \xi _{f}^{-}<\xi _{f}^{+}< \xi _{b}^{+}< \xi _{b}^{-} < L\). We can assume \(\xi _{f}^{-}=0\) without loss of generality. We can also assume \(u_{f}(0) \in \varPi _{f}^{-}\) and \(u_{b}(\xi _{b}^{+}) \in \varPi _{b}^{+}\) by suitable translations. \(A_{per}(\xi ;\lambda ,L)\) depend on u alone and is independent of other variables vwz. Also \(F(U,c,\varepsilon ,\alpha )\) has a symmetry \(F(-U,c,\varepsilon ,\alpha )=-F(U,c,\varepsilon ,\alpha )\). Therefore, we can conclude that \(\xi _{f}^{+} = \xi _{b}^{-}-\xi _{b}^{+}\) and \(u_{per}(\xi + \xi _{b}^{+}; L) = -u_{per}(\xi ; L)\). We now have

Lemma 4

There is a number \(\xi _\infty \) such that
$$\begin{aligned} \lim _{L \rightarrow \infty } \xi _{f}^{+}(L) = \lim _{L \rightarrow \infty } \left( \xi _{b}^{-}(L)-\xi _{b}^{+}(L) \right) = \xi _{\infty }. \end{aligned}$$
Moreover, \(\xi _{f}^{+}(L)-\xi _{b}^{+}(L)=L-\xi _{b}^{-}(L)\) holds, and \(L-\xi _{b}^{-}(L)\) tends to infinity as L goes to infinity.

Proof

It is clear if we take \(\xi _{\infty }\) as the time interval for the front heteroclinic orbit \(h_{f}(\xi )\) passes from \(\varPi _{f}^{-}\) through \(\varPi _{f}^{+}\). \(\square \)

Let us prepare the following lemma, which means that the eigenvalues of the linearized operator \({\mathcal {L}}_{per,L}\) accumulate to the essential spectrum \(\varSigma _{ess}\) as \(L \rightarrow \infty \). Theorem 2 is a immediate consequence of the following lemma because \(S_{per} \cap \{ \lambda \mid \mathop {\mathrm {Re}}\nolimits \lambda >0 \} \ne \emptyset \) by Lemmas 2 and 3 if \(\alpha \) is sufficiently small. Let \(\lambda _{0} \in S_{per}\) and take a disk \(B(\lambda _{0};\delta )\) centered at \(\lambda _{0}\) with radius \(\delta \).

Lemma 5

For arbitrary \(m \in {\mathbb {N}}\) and \(\delta >0\) there exists \(L_{*}>0\) such that the operator \({\mathcal {L}}_{per, L}\) has at least m eigenvalues in \(B(\lambda _0;\delta )\) for all \(L \ge L_{*}\).

This lemma follows from Lemmas 610 below.

Let us consider the extended ODE system of (19) to treat the boundary condition as a linear subspace.
$$\begin{aligned} \left\{ \begin{aligned} Y'&= A_{per}(\xi ;\lambda ,L)Y,\\ W'&= 0. \end{aligned} \right. \end{aligned}$$
(23)
Now, let us define a subspace of \({\mathbb {C}}^{8}\) by \(U_{per}=\{ (Y,Y) \mid Y \in {\mathbb {C}}^{4} \}\). It is clear that the fundamental solution matrix \(\bar{\varPhi }(\xi ,\zeta ;\lambda ,L)\) for (23) is given by \(\bar{\varPhi }(\xi ,\zeta ;\lambda ,L)=\varPhi (\xi ,\zeta ;\lambda ,L)+id_{4}\) where \(\varPhi (\xi ,\zeta ;\lambda ,L)\) is the fundamental solution matrix for (19). Here, \(id_{4}\) denotes \(\mathop {\mathrm {diag}}\nolimits \{ 1,1,1,1\}\). We can describe the boundary by the following lemma.

Lemma 6

\(\lambda \) is an eigenvalue of \({\mathcal {L}}_{per,L}\) if \(\bar{\varPhi }(L,0;\lambda ,L)U_{per}\cap U_{per} \ne 0\) holds.

The accumulation of eigenvalues occurs on the intervals \([\xi _{f}^{+},\xi _{b}^{+}]\) and \([\xi _{b}^{-},L]\) since \(\xi _{b}^{+}-\xi _{f}^{+} \rightarrow \infty \) and \(L - \xi _{b}^{-} \rightarrow \infty \) hold when \(L \rightarrow \infty \). Therefore, we consider the eigenvalue problem locally on the intervals \([\xi _{b}^{-},L]\) or \([\xi _{f}^{+},\xi _{b}^{+}]\) to see the accumulation of eigenvalues. If we write \(A_{per}(\xi ;\lambda ,L)=A_{0}(\lambda ) + B_{L}(\xi ;\lambda ,c)\), \(\Vert B_L \Vert \) is negligible by taking L large enough and \(\gamma \) small enough.

Therefore, (23) can be approximated by
$$\begin{aligned} \left\{ \begin{aligned} Y'&= A_{0}(\lambda )Y,\\ W'&= 0, \end{aligned} \right. \end{aligned}$$
(24)
on the both intervals.

Let \(\bar{E}^{c}(\lambda )\) be the generalized eigenspace corresponding to \(\mu ^{*}(\lambda )\) and 0. Then, \(\bar{E}^{c}(\lambda )\) and \(U_{per}\) are the general position with each other since \(\mu ^{*}(\lambda ) \ne 0\). We can say \(\bar{E}^{c}(\lambda ) + U_{per} ={\mathbb {C}}^{8}\) instead. Noticing that \(\bar{\varPhi }(\xi ,\zeta ;\lambda ,L)\)is a fundamental solution matrix, it holds that \(G_{i} + \bar{E}^{c}(\lambda )={\mathbb {C}}^{8}, \ i=1,2,3\) where \(G_{1}=\bar{\varPhi }(\xi _{f}^{+},0;\lambda ,L)U_{per}\), \(G_{2}=\bar{\varPhi }(\xi _{b}^{+},0;\lambda ,L)U_{per}\) and \(G_{3}=\bar{\varPhi }(\xi _{b}^{-},0;\lambda ,L)U_{per}\). Therefore eigenvalue problem of \({\mathcal {L}}_{per}\) can be considered as follows,

Lemma 7

\(\lambda \) is an eigenvalue of \({\mathcal {L}}_{per,L}\) if and only if \(\bar{\varPhi }(L,\xi _{b}^{-};\lambda ,L)G_{3} \cap U_{per} \ne 0\) holds.

It holds that \(\dim (\bar{E}^{c}(\lambda ) \cap G_{3})=1\) since \(\dim (\bar{E}^{c}(\lambda ) \cap U_{per})=1\). Now consider \({\mathbb {P}}(\bar{\varPhi }(L,\xi _{b}^{-};\lambda ,L)P_{3}) \), where \(P_{3}=\bar{E}^{c}(\lambda ) \cap G_{3}\). Notice that \(P(\xi ;\lambda )=\bar{\varPhi }(\xi ,\xi _{b}^{-};\lambda )G_{3} \cap \bar{E}^{c}(\lambda )\) is a 1 dimensional complex vector space in \(\bar{E}^{c}(\lambda )\). Then the differential equation which is induced on \({\mathbb {CP}}^{7}\) by (23) is controlled by the following restricted equation on \({\mathbb {P}} (\bar{E}^{c}(\lambda ) / E^{e}(\lambda )) \cong {\mathbb {CP}}^{1}\):
$$\begin{aligned} P'=\mathcal{P}(P,\xi ;\lambda ,L), \ P \in {\mathbb {CP}}^{1}. \end{aligned}$$
(25)
Here, \(E^{e}(\lambda )\)is the generalized eigenspace corresponding to 0, and \({\mathbb {P}} (\bar{E}^{c}(\lambda ) / E^{e}(\lambda ))\) is projectivization of the quotient vector space \(\bar{E}^{c}(\lambda ) / E^{e}(\lambda )\).
The above equation (25) can be described as
$$\begin{aligned} \eta ' = -\mu ^{*}(\lambda )\eta - \beta _{L}(\eta ,\theta _{1},\theta _{2},\theta _{3},\xi ,c;\lambda ), \end{aligned}$$
after an appropriate transformation. Here, \(\eta = q/p\) is an inhomogeneous coordinate on \(N_{+}=\{ [p:q] \mid p \ne 0 \} \subset {\mathbb {CP}}^{1}\) and \(\theta _{1}=Y_{1}/Y_{2}, \theta _{2}=Y_{3}/Y_{2},\theta _{3}=Y_{4}/Y_{2}\). It can be similarly described as
$$\begin{aligned} \eta ' = \mu ^{*}(\lambda )\eta + \beta _{L}(\eta ,\theta _{1},\theta _{2},\theta _{3},\xi ,c;\lambda ), \end{aligned}$$
by using an inhomogeneous coordinate \(\eta = p/q\) on \(N_{-}=\{ [p:q] \mid q \ne 0 \}\). If L is sufficiently large and \(\gamma \) is small then \(|\beta _{L}|\) is small. Let \(P(\xi ;\lambda )\) be a solution of (25) with \(P(\xi _{b}^{-};\lambda ) = G_{3} \cap \bar{E}^{c}(\lambda )\) and \(P_{per} = \bar{E}^{c}(\lambda ) \cap U_{per}\). Now we have,

Lemma 8

If \(P(L;\lambda ) =P_{per}\) then \(\lambda \) is an eigenvalue of \({\mathcal {L}}_{per}\).

We consider an ordinary differential equation which is induced on \({\mathbb {CP}}^{1}\) from (24):
$$\begin{aligned} P'=\mathcal{P}_{0}(P;\lambda ,L), \ P \in {\mathbb {CP}}^{1}. \end{aligned}$$
(26)
Let \(P_{0}(x;\lambda )\) be a solution of (26) with \(P_{0}(\xi _{b}^{-};\lambda ) \in G_{3}\). By taking the above inhomogeneous coordinate on \({\mathbb {CP}}^{1}\), we can express \(P_{0}(L;\lambda )=e^{-\mu ^{*}(L-\xi _{b}^{-})}G_{3}\) (or \(P_{0}(L;\lambda )=e^{\mu ^{*}(L-\xi _{b}^{-})}G_{3}\)).
Choose \(\delta > 0\) sufficiently small so that the following two conditions hold:
  1. 1.

    \(B(\lambda _{0};\delta ) \setminus S_{per}\) consists of a disjoint union of half disks \(B_{1}\) and \(B_{2}\), and \(\mathop {\mathrm {Re}}\nolimits \mu ^{*}(\lambda ) > 0\) for any \(\lambda \in B_{1}\) and \(\mathop {\mathrm {Re}}\nolimits \mu ^{*}(\lambda ) < 0\) for any \(\lambda \in B_{2}\).

     
  2. 2.

    \(\omega (\lambda )\) is monotone on \(S_{per}\).

     
Put \(\psi _{0,L}:=P_{0}(L;\cdot ):B(\lambda _{0};\delta ) \rightarrow {\mathbb {CP}}^{1}\). It is reasonable to expect that the following lemma holds by the representation of \(P_{0}(L;\lambda )\). The details of the proof are shown in [19] and [18].

Proposition 9

([18, 19]) Let \(N_{p}\) and \(N_{q}\) be neighborhoods of [1 : 0] and [0 : 1] satisfying \(N_{p}\cap N_{q} = \emptyset \) and \(P_{per} \in {\mathbb {CP}}^{1} / (N_{p} \cup N_{q})\). For any positive integer \(m \in {\mathbb {N}}\), there are \(\delta >0\) and \(L_{*}>0\) such that the following holds: for any \(L \ge L_{*}\) there exists a number \(m'(L) \ge m\) such that \(\psi _{0,L}:B(\lambda _{0};\delta ) \rightarrow {\mathbb {CP}}^{1} / (N_{p} \cup N_{q})\) is an \(m'(L)\)-covering map.

We put \(\psi _{L}:=P(L;\cdot ):B(\lambda _{0};\delta ) \rightarrow {\mathbb {CP}}^{1}\). Since \(\beta _{L}\) is small if L is sufficiently large and \(\gamma \) is small, the following lemma holds.

Lemma 10

There are \(\delta >0\) and \(L_{*}>0\) such that the following holds: for any \(L \ge L_{*}\) there is a number \(m'(L) \ge m\) such that \(\psi _{L}\) is also an \(m'(L)\)-covering map.

This completes the proof of Theorem 2.

Finally, we prove Theorem 3. We take \(\alpha \) sufficiently small so that \(S_{per} \cap \{ \lambda \mid \mathop {\mathrm {Re}}\nolimits \lambda >0 \} \ne \emptyset \). Eigenvalues of \(A_{0}(\lambda )\) has the following property.

Lemma 11

Let \(\lambda _{0}=\lambda (k_{*}) \in \varSigma _{ess} \cap i {\mathbb {R}}\) with \(k_{*}=\sqrt{1-\alpha u_{0}^{2}}\). Eigenvalues of \(A_{0}(\lambda _{0})\) consist of \(\mu ^{u}(\lambda _{0}),\mu ^{*}(\lambda _{0}),\mu ^{s}(\lambda _{0})\), and \(\mu ^{ss}(\lambda _{0})\) satisfying
$$\begin{aligned} \mathop {\mathrm {Re}}\nolimits \mu ^{u}(\lambda _{0})> \mathop {\mathrm {Re}}\nolimits \mu ^{*}(\lambda _{0})=0 > \mathop {\mathrm {Re}}\nolimits \mu ^{s}(\lambda _{0}) \ge \mathop {\mathrm {Re}}\nolimits \mu ^{ss}(\lambda _{0}). \end{aligned}$$

Proof

It is directly computed using the characteristic polynomial \(K_{\alpha }(\mu (\lambda _{0}))\) of \(A_{0}(\lambda _{0})\) and its derivative at \(\lambda _{0}\). \(\square \)

We again take a sufficiently small disk \(B(\lambda _0, \delta )\) and \(\lambda \in B(\lambda _0, \delta )\). Let \(E^{u}(\lambda )\) and \(E^{c}(\lambda )\) be the eigenspaces corresponding to \(\mu ^{u}(\lambda )\) and \(\mu ^{*}(\lambda )\), respectively, and let \(E^{s}(\lambda )\) be the generalized eigenspace corresponding to \(\mu ^{s}(\lambda )\) and \(\mu ^{ss}(\lambda )\). We then consider an equation which is induced on \({\mathbb {CP}}^{3}\) from (21):
$$\begin{aligned} Q'=\mathcal{Q}(Q,\xi ;\lambda ), \quad Q \in {\mathbb {CP}}^{3}. \end{aligned}$$
(27)
Notice that we have the same induced equation for both \(* = f\) and \(* = b\). Let \(Q_{u}(\xi ;\lambda )\) and \(Q_{c}(\xi ;\lambda )\) be solutions of (27) satisfying
$$\begin{aligned} \lim _{\xi \rightarrow -\infty }Q_{u}(\xi ;\lambda )=E^{u}(\lambda ),\quad \lim _{\xi \rightarrow -\infty }Q_{c}(\xi ;\lambda )=E^{c}(\lambda ), \end{aligned}$$
respectively. By Theorem 1, Lemma 11, and the symmetry of \(F(U,c,\varepsilon ,\alpha )\), the following lemma holds.

Lemma 12

\(Q_{u}(\xi ;\lambda _{0})\) and \(Q_{c}(\xi ;\lambda _{0})\) satisfy \(\lim _{\xi \rightarrow \infty }Q_{u}(\xi ;\lambda _{0}) \in E^{s}(\lambda )\) and \(\lim _{\xi \rightarrow \infty }Q_{c}(\xi ;\lambda _{0})=E^{c}(\lambda _{0})\), respectively.

We define \(\varphi _{1,\lambda }=\varPhi (\xi _{f}^{+},0;\lambda )\) and \(\varphi _{3,\lambda }=\varPhi (\xi _{b}^{-},\xi _{b}^{+};\lambda )\). Since \(h_{per}\) has the symmetry \(h_{per}(\xi + \xi _{b}^{+};c(L),L)=-h_{per}(\xi ;c(L),L)\), \(\varphi _{1,\lambda } = \varphi _{3,\lambda }\) holds. We also define \(\varphi _{2,\lambda ,L}=\varPhi (\xi _{b}^{+},\xi _{f}^{+};\lambda )\) and \(\varphi _{4,\lambda ,L}:=\varPhi (L,\xi _{b}^{-};\lambda )\). Similarly, \(\varphi _{2,\lambda ,L}=\varphi _{4,\lambda ,L}\) holds.

Let \(\hat{\varphi }_{i,\lambda }, \ i=1,3\) and \(\hat{\varphi }_{j,\lambda ,L}, \ j=2,4\) be maps which are induced on \({\mathbb {CP}}^{3}\) from \(\varphi _{i,\lambda } \ i=1,3\) and \(\varphi _{j,\lambda ,L}, \ j=2,4\), respectively. By Lemmas 4 and 12, \(\varphi _{1,\lambda }\) and \(\varphi _{3,\lambda }\) are determined by an independent of sufficiently large L. Therefore, we can realize the situation where \(\hat{\varphi }_{1,\lambda }(E^{c}(\lambda ))\) and \(\hat{\varphi }_{3,\lambda }(E^{c}(\lambda ))\) are arbitrarily close to \(E^{c}(\lambda )\) when \(\lambda = \lambda _{0}\), by taking L sufficiently large and \(\gamma \) sufficiently small. By the above argument, \(\hat{\varphi }_{2,\lambda ,L}(E^{c}(\lambda ))\) and \(\hat{\varphi }_{4,\lambda ,L}(E^{c}(\lambda ))\) are arbitrarily close to \(E^{c}(\lambda )\) when \(\xi _{b}^{+}-\xi _{f}^{+}=L-\xi _{b}^{-}=\pi n/\omega (\lambda _{0})\) for sufficiently large n. Therefore, \(\hat{\varphi }_{2,\lambda ,L_{n}} \circ \hat{\varphi }_{1,\lambda }(E^{c}(\lambda ))\) and \(\hat{\varphi }_{4,\lambda ,L_{n}} \circ \hat{\varphi }_{3,\lambda }(E^{c}(\lambda ))\) are arbitrarily close to \(E^{c}(\lambda )\) for sufficiently large n where \(L_{n}=\xi _{f}^{+} + (\xi _{b}^{-} - \xi _{b}^{+}) + 2\pi n /\omega (\lambda _{0})\). Moreover, it follows then, for any vector \(v(\lambda ) \in E^{c}(\lambda _{0})\) with \(v(\lambda ) \ne 0\), \(\hat{\varphi }_{1,\lambda _{0}}v=e^{i \omega (\lambda _{0})\xi _{f}^{+}}v + \text {h.o.t.}\) by the symmetry. This claim is a special case of [16, Eq. (5.2)]. We then define a 4-dimensional linear subspace \(U_{per}^{*}=\{ (-Y,Y) \mid Y \in {\mathbb {C}}^{n}\}\) and \(P_{per}^{*}=\bar{E}^{c}(\lambda ) \cap U_{per}^{*}\). From the above argument, we have two solutions \(Y_{e}\) and \(Y_{o}\) to (24) satisfying the following properties. That is, \(Y_{e}(0;\lambda _{0}), Y_{e}(\xi _{b}^{+};\lambda _{0})\), and \(Y_{e}(L_{n};\lambda _{0})\) are contained in \(U_{per}\) for an even n. On the other hand, \(Y_{o}(0;\lambda _{0})\) and \(Y_{o}(L_{n};\lambda _{0})\) are contained in \(U_{per}\) while \(Y_{o}(\xi _{b}^{+};\lambda _{0}) \in U_{per}^{*}\) for an odd n. Translating it into the terms of the projective space, we have two solutions \(P_{e}\) and \(P_{o}\) to (26) with the initial data \(P_{per}\) at \(\xi = 0\) satisfying the following properties. \(P_{e}\) corresponds to \(Y_{e}\) and satisfies \(P_{e}(\xi _{b}^{+};\lambda _{0}) = P_{e}(L_{n};\lambda _{0})=P_{per}\) for an even n. \(P_{o}\) corresponds to \(Y_{o}\) and satisfies \(P_{o}(L_{n};\lambda _{0})=P_{per}\) and \(P_{o}(\xi _{b}^{+};\lambda _{0})=P_{per}^{*}\) for an odd n. Let \(P(\xi ;\lambda )\) be a solution of (25) satisfying \(P(0;\lambda ) = P_{per}\). Although \(P(L_{n};\lambda _{0})\) is no longer equal to \(P_{per}\), we can clearly adjust \(\lambda \) and L so that \(P(L;\lambda ) = P_{per}\) holds for \((L, \lambda ) \approx (L_{n}, \lambda _{0})\) with \(\lambda \in i\mathbb {R}\setminus \{ 0 \}\).
Fig. 9

Schematic pictures of two solutions in \({\mathbb {CP}}^{1} \times [0,L]\). These solutions are characterized by the index n on \([\xi _{f}^{+}, \xi _{b}^{+}]\) and \([\xi _{b}^{-},L]\). a The first case : the index n is an even number. b The second case : the index n is an odd number

Summarizing the above arguments, there are complex conjugate eigenvalues of \({\mathcal {L}}_{per,L}\) in \(\mathrm {i}{\mathbb {R}}\) near by
$$\begin{aligned} L_{n}=2 \frac{\pi n}{\omega (\lambda _{0})} + 2 \xi _{\infty }. \end{aligned}$$
Adjusting n, we can take \(L_{\infty }\) and \(h_n\) so that (15) holds. Thus we obtain Theorem 3.

Remark 2

By using the solutions \(P_{e}\) and \(P_{o}\) to (26), we can distinguish to which type of torus bifurcation the critical eigenfunction leads. Indeed, if the index n is even, then the bifurcating rippling rectangular wave is anti-phase. On the other hand, an in-phase solution bifurcates if n is odd. Figure 9 illustrates these situations.

5 Discussion

We have demonstrated both theoretically and numerically that the torus bifurcation occurs successively on the branch of rectangular waves as the system size L increases. Since the curve of essential spectrum crosses the imaginary axis we can construct critical eigenfunctions which correspond to the “TRn” torus bifurcation. The profile of the critical eigenfunction consists basically of the sinusoidal part and the transition part along the front. This is because the rectangular wave stays close to the two steady states \(\pm u_0\) except the two transition regions. The sinusoidal part can be characterized by the eigenfunction corresponding to a pure imaginary eigenvalue for the eigenvalue problem about \(\pm u_0\). As mentioned in Remark 2, there are two types of critical eigenfunctions depending on their periods: L or L / 2. The type of bifurcating solution is in-phase when the period of the critical eigenfunction is exactly L. While on the other hand, anti-phase solutions bifurcate when the period is L / 2.

Suppose that \(\alpha = 0\). If the first torus bifurcation is supercritical, then the solution TR1 is stable. This is natural from the viewpoint of bifurcation theory. The other rippling rectangular waves are considered to be unstable if L is near the corresponding torus bifurcation point. However, when L is far from the bifurcation point, the rippling rectangular waves can be observed stably as a numerical solution of an initial value problem for (1). Indeed, Fig. 5 is obtained by solving an initial value problem numerically. This suggests that they are stable at such a parameter value. An unstable rippling rectangular wave should recover its stability by some bifurcations. When \(\alpha \) is varied from zero, as shown in Fig. 8, two torus bifurcation curves intersect. This means that two torus bifurcations occur simultaneously. This may be responsible for the secondary bifurcation at which the unstable rippling rectangular wave becomes stable. Therefore further study of whole bifurcation structure is necessary.

Notes

Acknowledgements

T.M. and T.O. are partly supported by JSPS KAKENHI Grant Number 16KT0023. T.M. is supported by JSPS KAKENHI Grant Number 16K17649, and T.O. is supported by JSPS KAKENHI Grant Number 17K05375. In this work, we used the computer of the MEXT Joint Usage / Research Center “Center for Mathematical Modeling and Applications”, Meiji University, Meiji Institute for Advanced Study of Mathematical Sciences (MIMS).

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Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  • Tomoyuki Miyaji
    • 1
    Email author
  • Toshiyuki Ogawa
    • 2
  • Ayuki Sekisaka
    • 3
  1. 1.Meiji Institute for Advanced Study of Mathematical SciencesMeiji UniversityTokyoJapan
  2. 2.School of Interdisciplinary Mathematical SciencesMeiji UniversityTokyoJapan
  3. 3.Meiji Institute for Advanced Study of Mathematical SciencesMeiji UniversityTokyoJapan

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