1 Introduction

Linear asymptotic stability and modulation behavior near periodic waves of the Korteweg–de Vries equation

Abstract.

We provide a detailed study of the dynamics obtained by linearizing the Korteweg–de Vries equation about one of its periodic traveling waves, a cnoidal wave. In a suitable sense, linearly analogous to space-modulated stability [14], we prove global-in-time bounded stability in any Sobolev space, and asymptotic stability of dispersive type. Furthermore, we provide both a leading-order description of the dynamics in terms of slow modulation of local parameters and asymptotic modulation systems and effective initial data for the evolution of those parameters. This requires a global-in-time study of the dynamics generated by a non normal operator with non constant coefficients. On the road we also prove estimates on oscillatory integrals particularly suitable to derive large-time asymptotic systems that could be of some general interest.

Keywords: periodic traveling waves; modulation systems; asymptotic stability; cnoidal waves ; Korteweg–de Vries equation; dispersive estimates; oscillatory integrals.

2010 MSC: 35B10, 35B35, 35P05, 35Q53, 37K45.

Research of L.Miguel Rodrigues was partially supported by the ANR project BoND ANR-13-BS01-0009-01.

1. Introduction

Substantial efforts have been recently devoted to complete a rather comprehensive theory analyzing the dynamics near periodic waves of dissipative systems of partial differential equations. We refer the reader to [14, 32, 33] for a thorough account of the available parabolic analysis. Comparatively, the parallel analysis of dispersive Hamiltonian systems seems still in its infancy. The goal of the present contribution to a general theory, still to come, is to show, on a case-study, how for the linearized dynamics one may completely recover the fine description of parabolic cases.

1.1. Preliminary observations

As a preliminary warning we strongly emphasize that we always consider waves as solutions of extended systems. This is perfectly common when dealing with asymptotically constant wave profiles — corresponding to solitary waves, kinks, shocks… — but is still rather unusual in the dynamical1 literature when focusing on periodic waves. We believe however that this is the right way to capture some features of wave propagation and in particular to incorporate the rich multi-scale space-time dynamics expected to occur near periodic waves. References on an alternative point of view focusing on bounded domains with periodic boundary conditions — and which has a long and successful history — may be found in [1, 19, 4]. Another related feature of our point of view is that we do not break invariance by translation by either focusing on a specific region of the space-time diagram or introducing weighted norms. Though the author does not know any implementation of these strategies in a periodic context2, for solitary waves and especially those of scalar equations the latter is now a well-established way to effectively bring Hamiltonian equations in a form as parabolic as possible, see [27, 19].

Here we restrict our attention to the Korteweg–de Vries equation as an archetype of dispersive Hamiltonian equation. It also comes with the key advantage of integrability. Indeed, whereas our ultimate goal is to show how to derive dynamical behavior from spectral information and thus to prove theorems where spectral properties are assumed, we believe that it is sounder to do so only after a sufficient number of spectral studies have gathered clear evidence of what is the best notion of spectral stability one may expect. Obviously the latter depends strongly on the nature of the background solution and on the class of system under consideration. We refer the reader to [32, 33] for detailed discussions of this question for periodic waves and especially of the notion of diffusive spectral stability that has slowly emerged as the relevant set of spectral conditions for periodic waves of dissipative systems. Yet, gathering either analytically or sometimes even numerically the relevant pieces of spectral information may often appear as a daunting task since in general profiles are not known explicitly and almost no a priori knowledge of even a part of the spectrum of relevant operators — that have variable coefficients that are not asymptotically constant – is available3. Moreover for the finer dynamical results one may need very precise description of the (critical part of the) spectrum, not usually included in classical spectral studies4. The reader may find in [15, 2] examples of spectral analyses required to apply the abstract parabolic theory. This is where, here, we use integrability to offer relatively elementary proofs of almost all relevant spectral claims and for the remaining ones, gathered in Assumption (A) below, a way to observe them by well-conditioned soft numerics. On this spectral side we strongly rely on the approach and results of [6].

Again, we are mostly interested in developing tools to study periodic waves of general dispersive equations so that the use of integrability is here restricted to obtaining relevant spectral information. However for the Korteweg–de Vries equation it is possible to use integrability by inverse scattering in a deeper way and derive large-time asymptotics directly at the nonlinear level, as proved in [23, 24]. Moreover the decay proved here at the linearized level turns out to be too slow to be used directly in any simple argument yielding a different proof of those nonlinear asymptotics for the Korteweg–de Vries equation. We stress however that the strategy of our proofs seems robust enough to be adapted to cases where one does expect to derive asymptotics for the nonlinear dynamics from bounds on the linearized evolution combined with a priori estimates. To the opinion of the author the analysis of a case where this occurs seems to be the next natural step in laying foundations for a general theory.

1.2. Bounded stability

After these preliminary warnings, we now start a precise account of results obtained in the present contribution. We choose the following form of the Korteweg–de Vries equation (KdV)

(1.1)

where is scalar and and denote time and space variable. A solution to (1.1) is called a periodic (uniformly traveling) wave if it has the form for some profile periodic of period one, some wavenumber and some time frequency . The corresponding phase velocity is then . We shall focus on the dynamics near a given wave with some fixed wavenumber and frequency and hence write (1.1) in the corresponding moving frame. Introducing through equation (1.1) becomes

(1.2)

so that is a steady solution of (1.2). Setting and performing a naive linearization yield where

(1.3)

We stress however that even if the linear decay obtained below were faster we would not expect to remain small at least in norms encoding some localization so that it is a priori unclear what is the role of this linearization in the description of the large-time dynamics. To be more specific let us recall that at the nonlinear level analyses of parabolic cases suggest that one should not expect to control , for some reasonable functional space but instead to bound

This encodes preservation of shapes but allows for a synchronization of phases by a sufficiently slow phase shift. That the corresponding notion of stability is indeed the relevant notion for periodic waves has slowly emerged along years. In [14] this notion — coined there as space-modulated stability — has been proved to be sharp for general parabolic systems. Moreover in [14] have also been identified what are necessary and sufficient cancellations in the structure of parabolic systems — called there phase uncoupling — to ensure usual orbital stability5 instead of space-modulated stability. Though the parabolic machinery obviously does not apply to (1.1) it is worth mentioning that (1.1) does not exhibit such null structures.

To unravel what is left of this at the linearized level, let us mimic the first steps of the natural strategy to prove space-modulated stability. Namely, one pick an initial datum and a couple such that

with sufficiently small and try to prove that there exists a corresponding solution such that there exists with small and . The first key observation is that in terms of the sought equation (1.2) takes the form

where is the operator defined above, and is nonlinear in and their derivatives, and, locally, at least quadratic. In particular neglecting terms expected to be at least quadratically small leaves

As readily observed this more involved point of view also leads to the consideration of the group generated by . But it also shows that linear space-modulated stability should be defined by requiring a control of

(1.4)

and not of . Therefore the following result should be interpreted as bounded linear stability of periodic waves of (1.1) in a space-modulated sense.

Theorem 1.1.

Bounded linear stability. For any , there exists such that for any such that and any time

We emphasize that the result is non trivial even for since is not a normal operator6 and is not expected to be a group of unitary transformations on any subspace of . Since coefficients of are not constant one should also notice that the proof requires an -by- analysis7. Indeed, in the reverse direction, there is now a large literature devoted to the analysis of growth rates of higher-order Sobolev norms for some classes of evolutions that are unitary on  — for instance those generated by linear Schrödinger equations including a (time-dependent) potential. Observe also that whereas the Korteweg-de Vries evolution supports an infinite number of conservations laws providing in other contexts a uniform control of Sobolev norms of solutions the kind of initial data considered here does not seem compatible with any form of integration of those conservation laws that would provide conservations of useful functionals so that they do not play any role in our analysis.

1.3. Asymptotic stability

We now turn to asymptotic linear stability. Since our focus is mostly on methodology we first recall some facts that are well-known to experts. Note first that the system at hand possesses classical real and Hamiltonian symmetries so that one already knows without any further knowledge that (marginally) spectrally stable8 waves come with -spectrum that lies on the imaginary axis. For the operator under study this may be extended to all Sobolev spaces by a rather general functional-analytic argument9 showing that the -spectrum does not depend on . In particular for none of the waves under consideration one expects exponential decay of , or of any part of it, in -norm. But once exponential decay has been ruled out corollaries of the Datko-Pazy theorem [25, Theorem 3.1.5 & Corollary 3.1.6] preclude any reasonable form of decay in -norm for .

In short, in the former paragraph we have recalled why for situations similar to those dealt with here, to exhibit decay one needs to leave the semi-group framework where initial data and solutions at later times are estimated in same norms. Specifically, here, the mechanism expected to yield some form of return to equilibrium is dispersive decay, in particular the trade-off is localization of the data against uniform decay of the solution. Therefore one aims at proving that the solution may be decomposed into a continuum of elementary blocks traveling with distinct velocities and that this effectively results into spreading and uniform decay10 of the solution. As expounded in Section 2, in the periodic context the existence of a continuous decomposition follows from the Bloch-wave decomposition11, where each block is parametrized by a Floquet exponent , combined with a spectral decomposition of Bloch symbol that provides the action of on the -part of the Floquet decomposition. Then we still need to know that velocities corresponding to the decomposition vary in a sufficiently non-degenerate way to yield dispersive spreading. This is the content of the following Assumption.

(A${}_{0}$)

We give a more precise account of Assumption (A${}_{0}$) in Section A. Despite the relatively explicit description of the spectrum of , the author has not been able to prove that this assumption does hold for any cnoidal wave. One may indeed prove in a rather straight-forward way that this condition holds in distinguished asymptotic regimes, that is, for large or for small eigenvalues. But out of these limits the fact that the explicit description does not provide directly a parametrization of the spectrum in terms of the Floquet exponent but rather a parametrization of both Floquet exponent and spectral parameter in terms of a third auxiliary variable, the spectral Lax parameter, leads any attempt to derive a closed-form for those derivatives to cumbersome expressions with signs not readily apparent. However based on easy well-conditioned numerics one may plot corresponding graphs. Experiments of the author — see Appendix A — based on eye inspection of those graphs clearly indicate that condition (A${}_{0}$) is always satisfied.

Theorem 1.2.

Asymptotic linear stability. For any cnoidal wave such that condition (A${}_{0}$) holds, there exists such that for any such that and any time such that12

In particular for any such wave, there exists such that for any time and any such that

As already alluded to above, establishing decay here involves the proof of global-in-time dispersive estimates for operators with variable coefficients — actually quite far from having constant coefficients. Needless to say that such form of results is quite unusual in the literature. The only related results the author is aware of are due to Cuccagna [7, 8] and Prill [28], and have been subsequently used to prove nonlinear results with sufficiently nice nonlinearities [9, 29]. A significant difference is that periodicity in their cases stem from the presence of a periodic potential, and that they linearize about the zero solution hence receive a self-adjoint operator, which extends the range of techniques available.

1.4. Slow modulation behavior

Actually numerical experiments suggests that a stronger version of condition (A${}_{0}$) holds, namely

(A)

The spectral point is associated with Floquet exponent . Thus through classical considerations on oscillatory integrals condition (A) leads to the fact that critical decay rate corresponds to the evolution of the spectral part of the initial data corresponding to small spectrum and small Floquet exponents, the remaining part of the solution decaying faster, at rate . It may then be expected that one could accurately describe the long-time evolution within a two-scale ansatz, essentially a periodic oscillation when looking at a fixed bounded domain but whose characteristics evolve in time and space on larger scales. Moreover the well-known fact that eigen modes corresponding to the spectral point are given in terms of variations at along the manifold of periodic traveling wave profile suggests that in a formal ansatz the local structure of oscillations could be captured by picking at each spatio-temporal point one neighboring periodic wave and slow evolutions would then result from a slow spatio-temporal motion along the manifold of periodic traveling waves. This is commonly referred to as a slow modulation behavior.

Our following results prove that this intuition is indeed correct at the linearized level we consider here and give a precise account of the large-time asymptotic behavior. To state it we first choose a parametrization of periodic traveling waves. Many choices are available in the present case, some of them being very explicit, others diagonalizing the first-order system formally driving at main order the slow evolution… We choose one of them that is not very explicit but both simple and known to be available essentially near any non degenerate wave of a system of partial differential equations. We refer the reader to [36, 18, 3] for a look at other possible parametrizations for the Korteweg–de Vries equation and to [5, Appendix B.2] or [32, Section 2.1] for a proof that our choice is still available in a broader context. In the context of the Korteweg–de Vries equation simple quadrature combined with reduction by symmetry shows that periodic traveling waves are smoothly given as with

Correspondingly the phase velocity is given as . Note in particular that for some average values , and some phase shift we have , and . By translating profiles with we may actually ensure . For the sake of writing convenience we shall do so from now on. In the following we shall also denote by the differential with respect to parameters . Now the validation at our linearized level of the slow modulation scenario takes the following simple form.

Theorem 1.3.

Slow modulation behavior. Assume that the cnoidal wave of parameters and phase shift zero is such that condition (A) holds. There exists such that for any such that there exist local parameters , and such that for any time with13

where is centered, , and are low-frequency and

We refer the reader to Section 3.1 for precise definitions of the intuitive notions of being centered or low-frequency. A few other comments are in order.

  1. The description implicitly contains the relation between local wavenumber and local phase shift that is familiar in spatio-temporal modulation theories.

  2. In the introduction we have chosen to state our results in a rather concise and abstract form. Yet our proof shows that may be chosen as explicit linear functions of .

  3. The fact that is centered for any expresses that the time dynamics can not create any global-in-space phase-shift. This may seem in strong contrast with what happens near traveling waves with localized variations, such as solitary waves or fronts, where the main effect of perturbations is usually captured by a global-in-space phase-shift that evolves in time. The heuristics is as follows. In any case perturbations affects solutions in a nearly local way. However for localized unimodal waves since at infinity the solution is approximately translation invariant and a single local shift effectively occurs the main effect may be described by a global-in-space phase-shift.

  4. Note carefully that does not decay so that even when is small one can not replace the first quantity estimated with a more nonlinear form

    This is consistent with the fact that the above form is not the right formula to undo the linearization expounded at the introduction of space-modulated norms. One correct formulation is that and, with

    and

    the quantity

    is bounded by a constant multiple of provided that is sufficiently small.

1.5. Averaged dynamics

The last thing we would like to do is to provide a large-time description of the dynamics of introduced in the foregoing theorem. In other words we would like to identify some equivalent averaged dynamics. Relevant effective equations may indeed be obtained as a correction to the famous first-order Whitham system, linearized about the constant parameters of our reference background wave

(1.5)

where is a -matrix and

is the averaged of the flux associated with conservation law for Benjamin’s impulse . A higher-order correction to the classical Whitham theory is required to describe accurately large-time behavior. Indeed one needs to reach a level of description accounting for dispersive effects, hence here the third order is the lowest relevant order of description.

Such a suitable system could actually be derived by arguing on formal grounds and those formal derivations may be thought as geometric optic expansions of WKB type, following from a higher-order version of the two-timing method of Whitham in the spirit of [26]. However to keep the analysis as tight as possible we here follow a different process expounded below.

The gain on going from the full scalar linear equation to system (1.5) is of averaging nature as has periodic coefficients while linear averaged systems are constant-coefficients systems and as such are expected to be much easier to understand directly. Yet as in classical homogenization problems the coefficients of reduced systems require averaging quantities depending on solutions of cell-problems. In particular has a quite daunting explicit form. It should be noted however that on one hand knowing that such reduction exists disregarding the specific form of the system already yields a wealth of information and that on the other hand if needed the coefficients involved may be computed numerically in a relatively simple way.

In any case the above-mentioned formal arguments do not yield any insight on effective initial data. Besides putting on sound mathematical grounds those formal arguments the main achievement of the following result is to provide equivalent initial data for averaged systems.

Theorem 1.4.

Averaged dynamics, third order. Assume that the cnoidal wave of parameters and phase shift zero is such that condition (A) holds. There exists such that for any such that there exists centered and low-frequency such that with

and for any such the local parameters of Theorem 1.3 may be chosen in such a way that for any time with

and

where denotes the solution operator for System 1.5.

Of course the point is that is negligible in front of in the large-time limit. Note also that while does not decay to zero in the large-time limit we do provide a description of up to eventually vanishing terms. This is a crucial achievement since creating phase shifts is indeed the leading effect of perturbations. It is important14 however to understand that whereas knowing phase shifts is in principle sufficient to construct a leading-order description of the original solution, the obtention of the the dynamical behavior of the phase itself requires a knowledge of all the modulation parameters. In particular even when one may enforce the time evolution will still create a significant phase shift.

The fact that in our statement the prescription of effective initial data for modulation equations is relatively simple in terms of is closely related to the fact that we choose to be low-frequency, which is consistent with a slow modulation scenario. It is actually possible to pick any such that and obtain equivalent statement where replaces but then effective initial data have a more complicated form that encode projection to slow phase shift. Explicitly in this case, in Theorems 1.4 and 1.5, should be replaced with

(1.6)

See [14, Remark 1.14] for a more detailed, related discussion.

In view of the decay rates obtained in Theorem 1.3 and the heuristics concerning orders of vanishing derivatives of spectral curves one may rightfully wonder whether there is a way to obtain a more precise description achieving remainders. It is indeed possible to reach this precision if one replaces the third order correction with a pseudo-differential one. Moreover one may achieve rates intermediate between and infinitely close to by replacing the third order correction with higher-order differential corrections. This is the content of our last main results.

However it seems hard to obtain those higher-order corrections by formal arguments of geometric optic type. Instead the higher-order systems may be obtained directly in a way that we explain now. We first make the following observations, to be obtained as corollaries of the proofs of our main results, that the first-order system

(1.7)

is strictly hyperbolic and that when diagonalizing the corresponding operator as one obtains first-order expansions of the three Floquet eigenvalues , , passing trough the origin, as . Now we claim that it is sufficient to include dispersion corrections

(1.8)

through

where is the th order Taylor expansion of near . By convention we also include the pseudo-differential case where by choosing as a smooth real-valued function that coincide with in a neighborhood of zero. For simplicity, in (1.5), we have also chosen .

Theorem 1.5.

Averaged dynamics, higher order. Assume that the cnoidal wave of parameters and phase shift zero is such that condition (A) holds.
Let be an odd integer larger than , or .

There exists and a cut-off function such that for any such that there exists centered and low-frequency such that with

and for any such the local parameters of Theorem 1.3 may be chosen in such a way that for any time with

and

where is the solution operator to System 1.8.

The foregoing construction of follows closely the classical construction of artificial viscosity system as large-time asymptotic equivalents to systems that are only parabolic in the hypocoercive sense of Kawashima. We refer the reader for instance to [13, Section 6], [31], [14, Appendix B] or [32, Appendix A] for a description of the latter. A notable difference however is that in the diffusive context higher-order expansions of dispersion relations beyond the second-order necessary to capture some dissipation does not provide any sharper description as the second-order expansion already provides the maximal rate compatible with a first-order expansion of eigenvectors.

We stress also that a significant difference with the third-order case dealt with in Theorem 1.4 is the necessity to introduce the low-frequency cut-off . This is due to the fact that for higher-order expansions one can not derive good dispersion properties for the full evolution from the mere knowledge of such behavior for the low-frequency part. This is somehow analogous to the fact that slow expansions of well-behaved parabolic systems may produce ill-posed systems.

At last one may also improve the description of the phase itself up to remainders. But this requires a suitably tailored refined effective initial data.

Theorem 1.6.

Averaged dynamics, sharpest description. Assume that the cnoidal wave of parameters and phase shift zero is such that condition (A) holds.
Let be an odd integer larger than , or .

There exists such that for any such that there exists a low-frequency such that the local parameters of Theorem 1.3 may be chosen in such a way that for any time with

and

Before entering into proofs of our main statements, to make those statements slightly more concrete let us summarize what we have learned at leading order from Theorems 1.3 and 1.4. At leading order the behavior of is captured by a linear modulation of phase and the phase shift is the antiderivative of the first component of a three-dimensional vector that is at leading-order a sum of three linear dispersive waves of Airy type, each one traveling with its own velocity. In particular, three scales coexist : the oscillation of the background wave at scale in , spatial separation of the three dispersive waves at linear hyperbolic scale , width of Airy waves of size . This is illustrated by direct simulations in Figure 1. To fully appreciate the figure, note that oscillatory Airy tail is on the left for the left-hand side and right-hand side dispersive waves and on the right for the middle one.

Figure 1. Three looks at the same time-evolution. The background wave has elliptic parameter hence period approximately . Initial data for the perturbation is the product of a sinus with a Gaussian. Dark lines start from the center of the Gaussian, and corresponds to linear group velocities.

1.6. Perspectives

For the linearized Korteweg–de Vries equation itself, besides the question of proving condition (A), still remains the question of providing a derivation of suitable modulation systems similar to (1.8), when , by formal arguments, either by using directly a geometric optic ansatz or by expanding the Hamiltonian energy.

Recall also that the decay proved here is too slow to be directly relevant at nonlinear level. From this, two natural follow-up questions arise:

  1. At the nonlinear level, for the Korteweg–de Vies equation, can we still provide a — more nonlinear! — slow modulation description of the asymptotic behavior obtained in [23, 24] ?

  2. Can we perform a similar linear analysis in another situation that could be carried to the nonlinear level ?

On the latter, natural candidates are to be found in dynamics near periodic plane waves of dispersive systems in sufficienty high dimension.

1.7. Structure of the paper

The remaining part of the paper is devoted to the proofs of foregoing theorems. In the next section we first recall some elements of Bloch analysis, extract from [6] detailed information on the spectrum of and derive from it some representations of the corresponding time-evolution. In particular we provide both a spectral decomposition of the evolution and its counterpart in terms of Green kernels. We also gather there spectral asymptotic expansions in singular limits where either Floquet eigenvalues converge to zero or go to . In the third section we prove Theorems 1.1 and 1.2, by using respectively the above-mentioned spectral and kernel representations. In the fourth section we achieve the proofs of remaining results. Those rely mostly on low Floquet/low eigenvalue expansions in the spirit of [14] combined with suitable oscillatory integral estimates. Proofs of the latter are given in Appendix B. We point out that though the subject is quite classical Appendix B, oriented towards derivation of asymptotically equivalent systems, could be of some general interest. In Appendix A we gather some numerical experiments supporting that Assumption (A) always holds.

2. Spectral preparation

2.1. Integral transform

We first recall how to decompose any function into a sum of functions that are simpler from the point of view of periodicity, namely

(2.1)

with each periodic of period one, that is

Such an inverse formula may be obtained by rewriting appropriately an inverse Fourier decomposition. For this purpose we introduce direct and inverse Fourier transforms, via

Then the adequate integral transform, called the Bloch transform or the Floquet-Bloch transform, may be defined by

(2.2)

The Poisson summation formula provides an alternative equivalent formula

As follows readily from (2.2), is a total isometry from to . Interpolating with triangle inequalities also yields Hausdorff-Young inequalities, for ,

where denotes conjugate Lebesgue exponent, .

We have introduced the Bloch transform so as to turn differential operators with periodic coefficients in multipliers with respect to the Floquet exponent . Indeed for as in (1.3) we have

where each acts on periodic functions as

On each has compact resolvent and it depends analytically on in the strong resolvent sense.

2.2. Spectrum of

Now we recall the content of [6, Theorem 7.1], slightly extended by using [6, Remark 4] and some extra functional-analytic arguments.

We have fixed a cnoidal wave profile to (1.1). Though such waves form a four-dimensional family one may use Galilean invariance and invariances by spacial translation and a suitable scaling to restrict the present discussion to a one-dimensional sub-family

where is an elliptic parameter15, , denotes the corresponding Jacobi elliptic cosine function and wavenumber is such that

Corresponding velocity is then given by .

For as above we set , and . Then for any couple , if and only if there exists such that

and

Moreover in this case is a simple eigenvalue of and an eigenfunction is provided by

while a solution of the formally dual problem

is given by

Normalization of and ensures all together a suitable form of bi-orthogonality, detailed below, convergence to trigonometric monomials in the limit and the absence of singularities on in the limit .

To carry out our Floquet analysis we shall use some consistent labeling of the spectrum of each . To this purpose we first observe that on both

are decreasing, respectively from to and from to . Therefore we may parametrize the part of the spectrum of arising from as , in a way that ensures ; the map is increasing when is endowed with alphabetical order, and odd; for any , ; and for any , . The structure of the spectrum related to is less obviously read on above formulas even though some pieces of in