Nonlinear Stability of Periodic Traveling Wave Solutions of the Generalized Korteweg–de Vries Equation

Johnson, Mathew A.

doi:10.1137/090752249

Cited by 62 publications

(135 citation statements)

References 21 publications

(76 reference statements)

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“…In the case of generalized KdV equations, for which α = 2 in (2.1) but the nonlinearity is arbitrary, the nondegeneracy of the linearization at a periodic traveling wave was shown in [Joh09], for instance, to be equivalent to that the wave amplitude not be a critical point of the period; the proof uses the Sturm-Liouville theory for ODEs. Furthermore it was verified in [Kwo89], among others, at solitary waves (in all dimensions).…”

Section: Nondegeneracy Of the Linearizationmentioning

confidence: 99%

“…An obvious approach toward Theorem 4.1 is to rerun the arguments in the proof in [GSS87] and derive stability criteria, analogous to (4.5); see [Joh09], for instance, where the last condition in (4.5) was suitably modified in the case of generalized KdV equations. However, it is in general difficult to count the number of negative eigenvalues in the periodic wave setting.…”

Section: Where (42)mentioning

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%

“…The nondegeneracy of the linearization proves a spectral condition, which plays a central role in the stability of traveling waves (see [Wei87,Lin08] among others) and the blowup (see [KMR11], for instance) for the related, time evolution equation, and therefore it is of independent interest. In the case of generalized KdV equations, the nondegeneracy at a periodic traveling wave was identified in [Joh09], for instance, with that the wave amplitude not be a critical point of the period. Furthermore it was verified in [Kwo89], among others, at solitary waves.…”

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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Section: Nondegeneracy Of the Linearizationmentioning

confidence: 99%

Section: Where (42)mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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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“…(which is (31), corresponding to the set of conditions found by Johnson in [Joh09]). Let us now consider the cubic NLS nonlinearity, whose numerical behavior differs.…”

Section: Benchmarksmentioning

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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Point Spectrum: Reduction to Finite-Rank Eigenvalue Problems

Kapitula

Promislow

2013

Applied Mathematical Sciences

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Nonlinear Stability of Periodic Traveling Wave Solutions of the Generalized Korteweg–de Vries Equation

Cited by 62 publications

References 21 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

Point Spectrum: Reduction to Finite-Rank Eigenvalue Problems

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