# Rippling rectangular waves for a modified Benney equation

- 442 Downloads

## 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

*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.

*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):

*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.

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. 2–4 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

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

### Theorem 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.If \(c=c_{\infty }(\varepsilon )\), then (11) has two heteroclinic solutions \(h_{f}(\xi )\) and \(h_{b}(\xi )\) satisfyingwhere \(\pm U_{0}(c_{\infty }(\varepsilon )) = (\pm u_{0}, 0, 0, 0)\) are equilibria for (11) with \(u_{0}=\sqrt{3c_{\infty }(\varepsilon )}\).$$\begin{aligned}&\lim _{\xi \rightarrow -\infty }h_{f}(\xi )=-U_{0}, \ \lim _{\xi \rightarrow +\infty }h_{f}(\xi )=U_{0},\\&\lim _{\xi \rightarrow -\infty }h_{b}(\xi )=U_{0}, \ \lim _{\xi \rightarrow +\infty }h_{b}(\xi )=-U_{0}, \end{aligned}$$
- 3.\(c(\varepsilon ,L)\) converges to \(c_{\infty }(\varepsilon )\) as \(L \rightarrow \infty \), and hence,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.$$\begin{aligned} \lim _{L \rightarrow \infty } d_{H}(\{h_{per}(\xi ;c(\varepsilon ,L),L) \} , \varGamma )=0, \end{aligned}$$(13)

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 \(c_{*}=c_{\infty }(\varepsilon )\).

### 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

*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 1–3.

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

*L*). Especially, we focus on the cases where \(L < 100\). In this section, we set parameters

*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.

*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 TR

*n*are also referred to as TR

*n*\((n = 1, 2, \dots 8)\).

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 | | # 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 |

*L*increases, except first two bifurcations.

- 1.
At \(t = 1.8\), there are two humps on the lower plateau and two hollows on the higher plateau.

- 2.
At \(t = 2.7\), an additional hump and hollow appear simultaneously from the interface between lower and higher plateaus.

- 3.
At \(t = 3.6\), the leftmost hump and hollow reach the cliff and disappear simultaneously.

- 4.
At \(t = 4.5\), the solution recovers a similar profile as \(t = 1.8\), while it is shifted to the left.

- 1.
At \(t = 1.6\), there are two humps and two hollows on the lower and higher plateaus, respectively.

- 2.
At \(t = 2.4\), an additional hump appears on the lower plateau.

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

*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.

## 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, \(u_{0}=\sqrt{3c_{\infty }(\varepsilon )}\).

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.

### Definition 1

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

### Proof

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

### Lemma 3

### Proof

*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 *v*, *w*, *z*. 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

*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 6–10 below.

### 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.

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.

*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 \({\mathcal {L}}_{per}\).

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

### 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

### Proof

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

### 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.

*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 \(\lambda \in i\mathbb {R}\setminus \{ 0 \}\).

*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 “TR*n*” 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).

## References

- 1.Balmforth, N.J., Ierley, G.R., Worthing, R.: Pulse dynamics in an unstable medium. SIAM J. Appl. Math.
**57**, 205–251 (1997)MathSciNetCrossRefzbMATHGoogle Scholar - 2.Carr, J., Chow, S.-N., Hale, J.K.: Abelian Integrals and Bifurcation Theory. J. Diff. Eqns.
**59**, 413–436 (1985)MathSciNetCrossRefzbMATHGoogle Scholar - 3.Cushman, R., Sanders, J.: A codimension two bifurcations with a third order Picard-Fucks equation. J. Diff. Eqns.
**59**, 243–256 (1985)CrossRefzbMATHGoogle Scholar - 4.Doedel, E.J., Oldeman, B.E.: AUTO07-P: continuation and bifurcation software for ordinary differential equations. http://indy.cs.concordia.ca/auto/. Accessed 8 Mar 2017
- 5.Fenichel, N.: Geometric singular perturbation theory for ordinary differential equations. J. Diff. Eqns.
**31**, 53–98 (1979)MathSciNetCrossRefzbMATHGoogle Scholar - 6.Gardner, R.: Spectral analysis of long wavelength periodic waves and applications. J. Math. Pures Appl.
**72**, 415–439 (1993)MathSciNetGoogle Scholar - 7.Govaerts, W.: Numerical methods for bifurcations of dynamical equilibria. Society for Industrial and Applied Mathematics Philadelphia, USA (2000)CrossRefzbMATHGoogle Scholar
- 8.Hochbruck, M., Ostermann, A.: Exponential integrators. Acta Numerica
**19**, 209–286 (2010)MathSciNetCrossRefzbMATHGoogle Scholar - 9.Hunter, J.D.: Matplotlib: A 2D graphics environment. Comput. Sci. Eng.
**9**, 90–95 (2007)CrossRefGoogle Scholar - 10.Jorba, À.: Numerical computation of the normal behaviour of invariant curves of \(n\)-dimensional maps. Nonlinearity
**14**, 943–976 (2001)MathSciNetCrossRefzbMATHGoogle Scholar - 11.Kawahara, T., Toh, S.: Pulse interactions in an unstable dissipativedispersive nonlinear system. Phys. Fluids
**31**, 2103–2111 (1988)MathSciNetCrossRefGoogle Scholar - 12.Komatsu, T.S., Sasa, S.: Kink soliton characterizing traffic congestion. Phys. Rev. E
**52**, 5574–5582 (1995)CrossRefGoogle Scholar - 13.Kuznetsov, YuA: Elements of applied bifurcation theory, 3rd edn. Springer-Verlag, New York (2004)CrossRefzbMATHGoogle Scholar
- 14.Montagne, R., Hernández-García, E., Amengual, A., San, M.: Miguel. Wound-up phase turbulence in the complex Ginzburg-Landau equation. Phys. Rev. E
**56**, 151–167 (1997)MathSciNetCrossRefGoogle Scholar - 15.Ogawa, T.: Traveling wave solutions to a perturbed Korteweg-de Vries equation. Hiroshima Math. J.
**24**, 401–422 (1994)MathSciNetzbMATHGoogle Scholar - 16.Sandstede, B., Scheel, A.: Absolute and convective instabilities of waves on unbounded and large bounded domains. Phys. D
**145**, 233–277 (2000)MathSciNetCrossRefzbMATHGoogle Scholar - 17.Schilder, F., Osinga, H.M., Vogt, W.: Continuation of quasi-periodic invariant tori. SIAM J. Appl. Dyn. Sys.
**4**, 459–488 (2005)MathSciNetCrossRefzbMATHGoogle Scholar - 18.Sekisaka, A.: Topological approach to the stability properties of traveling waves for one-dimensional reaction diffusion systems. Ph. D thesis, Tohoku University (2015)Google Scholar
- 19.Sekisaka, A.: The absolute spectrum revisited from a topological viewpoint, arXiv:1706.08305, https://arxiv.org/abs/1706.08305 (2017). Accessed 27 June 2017
- 20.Williams, T., Kelley, C.: gnuplot homepage, http://gnuplot.info/ (2017). Accessed 27 June 2017

## Copyright information

**Open Access**This 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.