An index theorem for the stability of periodic travelling waves of Korteweg–de Vries type

Bronski, Jared C.; Johnson, Mathew A.; Kapitula, Todd

doi:10.1017/s0308210510001216

Cited by 74 publications

(151 citation statements)

References 27 publications

(57 reference statements)

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“…The proof follows from the Fredholm alternative and may be found in [BH14, Lemma 6]; see [BJK11] in the case of generalized KdV equations. Here we merely hit the main points.…”

Section: Nondegeneracy Of the Linearizationmentioning

confidence: 99%

“…Moreover we relate the latter condition with the momentum and the mass as functions of Lagrange multipliers arising in the traveling wave equation, generalizing that in [BSS87], for instance, in the solitary wave setting. In the case of generalized KdV equations, i.e., m(ξ) = ξ 2 in (1.1), the nonlinear stability of a periodic traveling wave to same period perturbations was determined in [Joh09], for instance, through spectral conditions, which were expressed in terms of eigenvalues of the associated monodromy map (or the periodic Evans function); see also [AP07,APBS06,BJK11,DK10,DN11]. Confronted with nonlocal operators, however, spectral problems may be out of reach by Evans function techniques.…”

Section: Introductionmentioning

confidence: 99%

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Stability of Periodic Traveling Waves for Nonlinear Dispersive Equations

Hur¹,

Johnson²

2015

SIAM J. Math. Anal.

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Abstract. We study the stability and instability of periodic traveling waves for Korteweg-de Vries type equations with fractional dispersion and related, nonlinear dispersive equations. We show that a local constrained minimizer for a suitable variational problem is nonlinearly stable to period preserving perturbations, provided that the associated linearized operator enjoys a Jordan block structure. We then discuss when the linearized equation admits solutions exponentially growing in time.

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“…The proof follows from the Fredholm alternative and may be found in [BH14, Lemma 6]; see [BJK11] in the case of generalized KdV equations. Here we merely hit the main points.…”

Section: Nondegeneracy Of the Linearizationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Stability of Periodic Traveling Waves for Nonlinear Dispersive Equations

Hur¹,

Johnson²

2015

SIAM J. Math. Anal.

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“…Note however that our conditions in (s) are slightly more general than the prescription of ((s1) or (s2)), in that they do not require θ λλ θ µµ − θ 2 λµ = 0. In the special case when κ is constant, Theorem 5 was essentially already known in the case θ λλ θ µµ − θ 2 λµ = 0, and proved in a slightly different manner in [BJK11]. Indeed, orbital stability with respect to co-periodic perturbations is essentially a consequence of [BJK11, Theorem 1], under the assumption…”

Section: So Our Main Assumptions (H0)-(h1)-(h2)-(h3)mentioning

confidence: 98%

“…, is a Sturm-Liouville operator with periodic coefficients, the main argument in the proof of Lemma 2 relies on Sturm's oscillation theorem, as for instance in Lemma 1 in [BJK11], which concerns the case of a constant κ. The detailed proof is postponed to Appendix B.…”

Section: So Our Main Assumptions (H0)-(h1)-(h2)-(h3)mentioning

confidence: 99%

Co-periodic stability of periodic waves in some Hamiltonian PDEs

2016

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The stability theory of periodic traveling waves is much less advanced than for solitary waves, which were first studied by Boussinesq and have received a lot of attention in the last decades. In particular, despite recent breakthroughs regarding periodic waves in reaction-diffusion equations and viscous systems of conservation laws [Johnson-Noble-Rodrigues-Zumbrun, Invent math (2014)], the stability of periodic traveling wave solutions to dispersive PDEs with respect to 'arbitrary' perturbations is still widely open in the absence of a dissipation mechanism. The focus is put here on co-periodic stability of periodic waves, that is, stability with respect to perturbations of the same period as the wave, for KdV-like systems of one-dimensional Hamiltonian PDEs. Fairly general nonlinearities are allowed in these systems, so as to include various models of mathematical physics, and this precludes complete integrability techniques. Stability criteria are derived and investigated first in a general abstract framework, and then applied to three basic examples that are very closely related, and ubiquitous in mathematical physics, namely, a quasilinear version of the generalized Korteweg-de Vries equation (qKdV), and the Euler-Korteweg system in both Eulerian coordinates (EKE) and in mass Lagrangian coordinates (EKL). Those criteria consist of a necessary condition for spectral stability, and of a sufficient condition for orbital stability. Both are expressed in terms of a single function, the abbreviated action integral along the orbits of waves in the phase plane, which is the counterpart of the solitary waves moment of instability introduced by Boussinesq. However, the resulting criteria are more complicated for periodic waves because they have more degrees of freedom than solitary waves, so that the action is a function of N + 2 variables for a system of N PDEs, while the moment of instability is a function of the wave speed only once the endstate of the solitary wave is fixed. Regarding solitary waves, the celebrated Grillakis-Shatah-Strauss stability criteria amount to looking for the sign of the second derivative of the moment of instability with respect to the wave speed. For periodic waves, stability criteria involve all the second order, partial derivatives of the action. This had already been pointed out by various authors for some specific equations, in particular the generalized Korteweg-de Vries equation -which is special case of (qKdV) -but not from a general point of view, up to the authors' knowledge. The most striking results obtained here can be summarized as: an odd value for the difference between N and the negative signature of the Hessian of the action implies spectral instability, whereas a negative signature of the same Hessian being equal to N implies orbital stability. Furthermore, it is shown that, when applied to the Euler-Korteweg system, this approach yields several interesting connexions between (EKE), (EKL), and (qKdV). More precisely, (EKE) and (EKL) share the same abbreviated a...

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“…That is, given (a 0 , E 0 , c 0 ) ∈ D with c 0 > 0, we assume that the Jacobian {T, M, P } a,E,c is non-zero. While these re-parametrization conditions may seem obscure, the non-vanishing of these Jacobians has been seen to be essential in both the spectral and non-linear stability analysis of periodic gKdV waves in [BrJ,J1,BrJK]. In particular, these Jacobians have been computed in [BrJK] for several power-law nonlinearities and, in the cases considered, has been shown to be generically non-zero.…”

Section: Preliminariesmentioning

confidence: 99%

On the modulation equations and stability of periodic generalized Korteweg–de Vries waves via Bloch decompositions

Johnson¹,

Zumbrun²,

Bronski³

2010

Physica D: Nonlinear Phenomena

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In this paper, we complement recent results of Bronski and Johnson and of Johnson and Zumbrun concerning the modulational stability of spatially periodic traveling wave solutions of the generalized Korteweg-de Vries equation. In this previous work it was shown by rigorous Evans function calculations that the formal slow modulation approximation resulting in the Whitham system accurately describes the spectral stability to long wavelength perturbations. Here, we reproduce this result without reference to the Evans function by using direct Bloch-expansion methods and spectral perturbation analysis. This approach has the advantage of applying also in the more general multiperiodic setting where no conveniently computable Evans function is yet devised. In particular, we complement the picture of modulational stability described by Bronski and Johnson by analyzing the projectors onto the total eigenspace bifurcating from the origin in a neighborhood of the origin and zero Floquet parameter. We show the resulting linear system is equivalent, to leading order and up to conjugation, to the Whitham system and that, consequently, the characteristic polynomial of this system agrees (to leading order) with the linearized dispersion relation derived through Evans function calculation.

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An index theorem for the stability of periodic travelling waves of Korteweg–de Vries type

Cited by 74 publications

References 27 publications

Stability of Periodic Traveling Waves for Nonlinear Dispersive Equations

Stability of Periodic Traveling Waves for Nonlinear Dispersive Equations

Co-periodic stability of periodic waves in some Hamiltonian PDEs

On the modulation equations and stability of periodic generalized Korteweg–de Vries waves via Bloch decompositions

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