Subharmonic Dynamics of Wave Trains in the Korteweg-de Vries / Kuramoto-Sivashinsky Equation

In this work, we consider the local dynamics of wave trains, i.e., periodic traveling wave solutions, of the Korteweg-de Vries / Kuramoto-Sivashinsky (KdV/KS) equation

where $x,t\in{\mathbb{R}}$. Here, $\varepsilon\geq 0$ and $\delta>0$ are modeling parameters which may, without loss of generality, be chosen such that $\varepsilon^{2}+\delta^{2}=1$: see Remark 1.1 below. The equation (1.1) is known to be a canonical model for pattern formation that has been used to describe many applications including plasma instabilities, turbulence in reaction diffusion equations, flame front propagation, and nonlinear wave dynamics in fluid mechanics [20, 21, 7, 15, 16]. In the case $\varepsilon=0$ and $\delta=1$, equation (1.1) becomes the classical Kuramoto-Sivashinsky equation, which is known to be a generic equation for chaotic dynamics, and there is a very large literature on these solutions, their bifurcations and period doubling cascades, and their stability: see, for example, [9] and reference therein. In this $\varepsilon=0$ case, (1.1) is also known to model thin film dynamics down a completely vertical wall [6]. When the angle of the wall is decreased from vertical, however, additional dispersive effects are present [9] and modeled by $\varepsilon>0$, and in the “flat” limit where the inclined wall becomes horizontal one recovers the completely integrable Korteweg-de Vries equation, corresponding here to $\varepsilon=1$ and $\delta=0$. Thus, in the context of inclined thin film flow one can consider the general model (1.1) as interpolating between the vertical wall $(\varepsilon,\delta)=(0,1)$ and the “flat” limit $(\varepsilon,\delta)\to(1^{-},0^{+})$. For more information on the connection to thin film dynamics, as well as its derivation in this context from the viscous shallow water equations or the full Navier Stokes equations, see [24, 25].

In this work, we are interested in understanding the stability and long-time dynamics of periodic traveling wave solutions of (1.1) to specific classes of perturbations. To begin our discussion, we briefly discuss the existence theory for periodic solutions of (1.1). First, note that traveling wave solutions of (1.1) correspond to solutions of the form $u(x,t)=\bar{u}(x-ct)$, where $\bar{u}$ necessarily satisfies the profile ODE

The existence and local structure of periodic solutions of (1.2) has been studied extensively by several other authors. For general values of $(\varepsilon,\delta)$, an elementary Hopf bifurcation analysis [3] shows the existence of a $3$-parameter family of asymptotically small amplitude periodic traveling wave solutions of (1.2) which, up to translation, can be parametrized by the wave speed $c$ and the period $T$. In the “classical KS” limit $|\varepsilon|\ll 1$, one can likewise use normal form analysis to establish a similar existence result [9], while the full bifurcation picture for the KS equation ($\varepsilon=0$) is known to be extremely complicated: see, for example, [14], where the authors prove the existence of a Shi’lnikov bifurcation which leads to cascades of period doubling, period multiplying $k$-bifurcations and oscillatory homoclinic orbits as the period is increased, as well as the numerical bifurcation study in [4]. For a summary and more details, see [3, 11].

Using the above existence studies as motivation, and closely following the work in [3], we make the following assumption regarding the existence of periodic solutions of (1.2), as well as the structure of the local manifold of periodic solutions.

As stated above, our main goal is to study the stability and dynamics of periodic traveling wave solutions of (1.1) to specific classes of perturbations. Previously, there has been much work regarding both the spectral and nonlinear stability of such periodic solutions when subjected to localized perturbations, i.e., perturbations that are integrable on the line [3, 2, 11], as well as their dynamics under slow-modulations [17]. In these works, it is found that for each admissible pair $(\varepsilon,\delta)$ in (1.1) there exist periodic traveling wave solutions which are spectrally and nonlinearly stable to localized perturbations. Here, however, we study the stability and dynamics of $T$-periodic traveling wave solutions of (1.1) when subject to so-called subharmonic perturbations, i.e., $NT$-periodic perturbations for some $N\in{\mathbb{N}}$. Before we continue, we note that an extremely important feature of (1.1), which we utilize heavily in our forthcoming analysis, is the presence of a Galilean symmetry. In particular, if $u(x,t)$ is a solution of (1.1), then so is the function

for any $c\in{\mathbb{R}}$. Thanks to this Galilean invariance, as well as the translational invariance of (1.1), it follows that the stability of a particular wave only depends on one parameter: namely, the period $T$. Thus, when discussing the stability of periodic traveling wave solutions of the KdV-KS equation (1.1), we identify all waves of a particular period. Furthermore, note we can view (1.3) as coupling between the wave speed and the mass of the wave, defined via

which is readily seen to be a conserved quantity of (1.1) due to the conservative structure.

Now, suppose that $\bar{u}(x)$ is a $T$-periodic solution of (1.2) with wave speed $c$, and note the linearization of (1.1) (in the appropriate co-moving frame) about $\bar{u}$ is given by the operator

Since we are considering subharmonic perturbations of $\bar{u}$, i.e., perturbations with period $NT$ for some $N\in{\mathbb{N}}$, here we consider $\mathcal{L}$ as a closed, densely defined linear operator acting on $L^{2}_{\rm per}(0,NT)$ with $T$-periodic coefficients. To describe the spectrum of $\mathcal{L}$ acting on $L^{2}_{\rm per}(0,NT)$, first observe that one always has that

so that $\lambda=0$ is an eigenvalue of $\mathcal{L}$ with algebraic multiplicity at least two and a Jordan chain of height at least one. With this in mind, and following the works [19, 18, 10, 13], we introduce the notion of spectral stability used throughout this work.

We note that in [3] it was shown that for every admissible pair of modeling parameters $(\varepsilon,\delta)$ there exists a range of periods for which all periodic traveling wave solutions of (1.1) with those period are diffusively spectrally stable: see Figure 1 and also [1, 5]. In the context of the KdV/KS equation (1.1), it is known that diffusively spectrally stable periodic traveling wave solutions (coupled with an additional non-degeneracy hypothesis, see Assumption 1.5 below) are nonlinearly stable to localized perturbations, in the sense that localized perturbations of such a periodic waves $\bar{u}$ converges to spatially localized phase modulations of $\bar{u}$. Specifically, given such a periodic traveling wave $\bar{u}$ and initial data

then for large time the associated solution $u(x,t)$ satisfies

for some function $\Psi(\cdot,t)\in W^{5,\infty}({\mathbb{R}})$ which behaves essentially like a finite sum of (small) error functions: see [3]. As we will see below, the implication of diffusive spectral stability will be that such a $T$-periodic traveling wave is necessarily spectrally stable to perturbations in $L^{2}_{\rm per}(0,NT)$ for every $N\in{\mathbb{N}}$.

Before continuing to state our main result, we introduce an additional non-degeneracy hypothesis. As we will see in Lemma 2.1 below, Assumption 1.2, along with the diffusive spectral stability assumption, implies that for $|\xi|$ small the Bloch operators $\mathcal{L}_{\xi}$ have two eigenvalues near the origin which expand as

for $|\xi|\ll 1$. Following the previous work [3, 10], we make the following additional non-degeneracy hypothesis:

By standard spectral perturbation theory, Assumption 1.5 ensures the analyticity of the functions $\lambda_{j}(\cdot)$ in (1.5): again, see Lemma 2.1 below. Furthermore, it is known that the coefficients $a_{j}$ are the characteristics of an associated Whitham averaged system, formally governing slowly modulated periodic solutions of (1.1). Consequently, Assumption 1.5 corresponds to strict hyperbolicity of the Whitham averaged system. See [17] for more details. Note also that all the diffusively spectrally stable periodic traveling wave solutions in Figure 1 were seen numerically to satisfy Assumption 1.5.

We now begin our discussion of our main results concerning the dynamics of $T$-periodic, diffusively spectrally stable traveling wave solutions of (1.1) when subjected to subharmonic perturbations. First, in Section 2.1 we will see that the spectrum of $\mathcal{L}$ acting on $L^{2}_{\rm per}(0,NT)$ is given by the union of the (necessarily discrete) spectrum of the corresponding Bloch operators $\mathcal{L}_{\xi}$ acting in $L^{2}_{\rm per}(0,T)$ for the discrete (finite) subset of $\xi\in[-\pi/T,\pi/T)$ such that $e^{i\xi NT}=1$. Thus, such a traveling wave $\bar{u}$ is necessarily spectrally stable to all subharmonic perturbations. In particular, for each $N\in{\mathbb{N}}$ there exists a constant $\delta_{N}>0$ such that

Since $\mathcal{L}$ is clearly sectorial, it is easy to show that for $N\in{\mathbb{N}}$ fixed and $\delta\in(0,\delta_{N})$ there exists a constant $C_{\delta}>0$ such that

for all $f\in L^{2}_{N}$, where here $\mathcal{P}_{N}$ denotes the projection of $L^{2}_{N}$ onto the $NT$-periodic generalized kernel of $\mathcal{L}$. Equipped with this linear estimate and exploiting the Galilean invariance (1.3), the following nonlinear stability result was established in [22, 23] for the case $N=1$ and is easily extended to general $N\in{\mathbb{N}}$.

The above result establishes the nonlinear asymptotic stability of the Galilean family associated to $\bar{u}$, showing that nearby $NT$-periodic solutions will, up to a spatial translation, asymptotically converge to a member of the Galilean family of $\bar{u}$. However, Proposition 1.6 lacks uniformity in $N$ in two important ways. Specifically, both the exponential rate of decay $\delta$ and the allowable size of initial perturbations $\varepsilon=\varepsilon_{\delta}$ are controlled by the size of the spectral gap $\delta_{N}>0$. Since it is known that $\delta_{N}\to 0$ as $N\to\infty$, it follows that both $\delta$ and $\varepsilon$ in Proposition 1.6 must necessarily tend to zero as $N\to\infty$. Of course, from a practical level it would be preferable to develop a theory which, for a given background wave $\bar{u}$, yielded a fixed size for the theoretically prescribed domain of attraction as well as a uniform (in $N$) rate of decay of initial perturbations. The fact that this is possible is precisely our main result.

The main difficulty in establishing the proof of Theorem 1.7 appears at the linear level. Specifically, one must develop a strategy to handle the accumulation of $NT$-periodic eigenvalues near $\lambda=0$ for $N\gg 1$ in a uniform way. In the proof of Proposition 1.6, this accumulation is handled by enclosing the origin in the spectral plane in an small ball $B(0,r_{N})$ where the $N$-dependent radius is chosen so that

and defining the associate Riesz spectral projection

onto the $NT$-periodic generalized kernel of $\mathcal{L}$. One then decomposes the linear solution operator (semigroup) as

and uses the exponential bound (1.6) to establish the result. The lack of uniformity in Proposition 1.6, however, stems from the fact that the radius $r_{N}$ of the ball used in the Riesz projection (1.9) must necessarily tend to zero as $N\to\infty$ in order to maintain the spectral decomposition (1.8).

To establish the uniformity in Theorem 1.7, however, we cannot allow the radius of the ball about the origin to shrink. Instead, we define a ball $B(0,r)$ with an $N$-independent radius and define the associated Riesz spectral projection associated to the total generalized eigenspace corresponding to eigenvalues interior to the ball. Naturally, the dimension of this total eigenspace is tending to infinity as $N\to\infty$, and we must work to establish uniform in $N$ decay estimates associated with the induced decomposition of the semigroup. This methodology was first introduced, at the linear level, by the authors and collaborators in [8] and was further extended, in the context of reaction-diffusion systems, to the full nonlinear level in [12], and is closely modeled off of the known stability theory for localized perturbations [10]. In the present work, the additional complication (compared to these previous subharmonic works) is the presence of a non-trivial Jordan block (1.4) which, in turn, yields slower uniform decay rates of the associated linear semigroups as compared to those rates in the reaction-diffusion context.

Finally, we make a few remarks concerning the connection between Proposition 1.6 and Theorem 1.7. The analysis in Section 4 shows that the modulation function $\widetilde{\psi}$ in Theorem 1.7 can be decomposed as

where here $\psi(\cdot,t)\in W^{5,\infty}_{N}$ satisfies

for some constant $C>0$ independent of $N$. Further, while our methods fail to directly yield convergence¹¹1This is due to the presence of a non-trivial Jordan block associated with the co-periodic Bloch opertaor $\mathcal{L}_{0}$, leading to slower linear decay rates in Section 3 when compared to, for example, the reaction-diffusion context where no such Jordan block is present (see [12] for details). of $\gamma(t)$ as $t\to\infty$, one immediate observation is that for each fixed $N\in{\mathbb{N}}$ and $NT$-periodic solution $u(x,t)$ of (1.1) with initial data sufficiently close (in $L^{1}_{N}\cap H^{5}_{N}$) to $\bar{u}$, then using the notation from both Proposition 1.6 and Theorem 1.7 we have

for some ($N$-dependent) constant $C_{N}>0$. This leads one to expect that, under relatively generic circumstances, for each fixed $N$ one should have

as $t\to\infty$. Understanding this convergence rigorously is an interesting remaining problem from our analysis.

The outline of this paper is as follows. In Section 2 we briefly review the application of Floquet-Bloch theory to subharmonic perturbations and collect several properties of the Bloch operators and their associated semigroups. We further establish basic “high-frequency” decay properties of the Bloch semigroups that arise as consequences of the diffusive spectral stability assumption. In Section 3 we provide a delicate decomposition of the semigroup $e^{\mathcal{L}t}$ acting on $L^{2}_{\rm per}(0,NT)$ which will yield polynomial decay rates on the semigroup which are uniform in $N$: see Proposition 3.1. This linear decomposition then motivates in Section 4 a nonlinear decomposition of a small $L^{2}_{\rm per}(0,NT)$ neighborhood of the underlying diffusively stable $T$-periodic background wave $\bar{u}$. Equipped with this nonlinear decomposition, along with the linear estimates from Section 3, we apply a nonlinear iteration scheme to corresponding system of perturbation equations, yielding a proof of our main result Theorem 1.7 above.

Here, we review several preliminary analytical results. We begin with a brief review of the application of Floquet-Bloch theory to subharmonic perturbations, and then use this to establish some elementary semigroup estimates for the associated Bloch operators. For notational convenience, for each $N\in{\mathbb{N}}$, $p\geq 1$, and fixed period $T>0$ we introduce the notation

and similarly with all associated Sobolev spaces.

The goal of this section is to obtain decay estimates on the semigroup $e^{\mathcal{L}t}$ acting on $L^{2}_{N}$ which are uniform in $N$. To this end, we use (2.4) to study the action of $e^{\mathcal{L}t}$ on $L^{2}_{N}$ in terms of the associated Bloch operators. Naturally, by Lemma 2.1 we expect for each fixed $N\in{\mathbb{N}}$ that the long-time behavior of $e^{\mathcal{L}t}$ is dominated by the generalized kernel of $\mathcal{L}$. As discussed in the introduction, the main difficulty in obtaining such a result while maintaining uniformity in $N$ is the accumulation of $NT$-periodic eigenvalues near the origin as $N\to\infty$. This is overcome through a delicate decomposition of $e^{\mathcal{L}t}$, separating the action into appropriate critical (i.e., corresponding to spectrum accumulating near the origin) and non-critical frequency components. This methodology is heavily influenced by the corresponding decomposition used in the case of localized perturbations of periodic waves: see, for example, [10]. Note, further, that while our basic strategy is similar to the recent subharmonic analysis of reaction diffusion equations conducted in [12], the new challenge here is the fact that the linearized operator $\mathcal{L}$ now has two spectral curves which pass through the origin, a reflection of the non-trivial Jordan structure in (1.4).

To begin, let $\xi_{1}\in(0,\pi/T)$ be defined as in Lemma 2.1 and let $\rho$ be a smooth cutoff function satisfying $\rho(\xi)=1$ for $|\xi|<\xi_{1}/2$ and $\rho(\xi)=0$ for $|\xi|>\xi_{1}$. Given a function $v\in L^{2}_{N}$, we then use (2.4) to decompose the action of $e^{\mathcal{L}t}$ on $v$ into high-frequency and low-frequency components via

Using Proposition 2.2(i), it follows that there exist constants $C,\eta>0$, both independent of $N$, such that

which, by the subharmonic Parseval identity (2.2), implies the exponential estimate

on the high-frequency component of the solution operator.

For the low-frequency component, define for each $|\xi|<\xi_{0}$ the rank-two spectral projection onto the critical modes of $\mathcal{L}_{\xi}$ by

Note we also have the alternate representation formula

which more explicitly demonstrates the singularity as $\xi\to 0$ of the eigenprojection. Using this, the low-frequency operator $S_{lf,N}(t)$ can be further decomposed into the contribution from the critical modes near $(\lambda,\xi)=(0,0)$ and the contribution from the low-frequency spectrum bounded away from $\xi=0$ via