Periodic finite-genus solutions of the KdV equation are orbitally stable

Nivala, Michael; Deconinck, Bernard

doi:10.1016/j.physd.2010.03.005

Cited by 23 publications

(32 citation statements)

References 45 publications

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“…The presence of instabilities in shallow water may be surprising, especially in the face of evidence from various asymptotic models such as the Korteweg-de Vries equation (see Bottman &Deconinck 2009 andNivala & and the Benjamin-Bona-Mahoney equation (see Haragus 2008) among others, demonstrating the stability of periodic waves in the context of these asymptotic models describing waves in shallow water. Those equations are all a consequence of long-wave assumptions, and they are not expected to capture the instabilities represented by the bubble eigenvalues.…”

Section: Discussionmentioning

confidence: 99%

The instability of periodic surface gravity waves

2011

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Euler's equations describe the dynamics of gravity waves on the surface of an ideal fluid with arbitrary depth. In this paper, we discuss the stability of periodic travelling wave solutions to the full set of nonlinear equations via a non-local formulation of the water wave problem, modified from that of Ablowitz, Fokas & Musslimani (J. Fluid Mech., vol. 562, 2006, p. 313), restricted to a one-dimensional surface. Transforming the non-local formulation to a travelling coordinate frame, we obtain a new formulation for the stationary solutions in the travelling reference frame as a single equation for the surface in physical coordinates. We demonstrate that this equation can be used to numerically determine non-trivial travelling wave solutions by exploiting the bifurcation structure of this new equation. Specifically, we use the continuous dependence of the amplitude of the solutions on their propagation speed. Finally, we numerically examine the spectral stability of the periodic travelling wave solutions by extending Fourier-Floquet analysis to apply to the associated linear non-local problem. In addition to presenting the full spectrum of this linear stability problem, we recover past well-known results such as the Benjamin-Feir instability for waves in deep water. In shallow water, we find different instabilities. These shallow water instabilities are critically related to the wavelength of the perturbation and are difficult to find numerically. To address this problem, we propose a strategy to estimate a priori the location in the complex plane of the eigenvalues associated with the instability.Key words: instability, surface gravity waves IntroductionEuler's equations for fluid motion are a set of nonlinear equations that describe surface gravity and capillary waves on an ideal fluid with arbitrary depth. A large class of commonly used mathematical models for ocean waves have been derived from these equations as approximations. Examples include the Korteweg-deVries equation (in the case of shallow water) and the nonlinear Schrödinger equation (in the case of deep water). While we can analytically determine solutions to some of the approximate equations, analytical solutions to the full set of Euler equations are unknown. Thus, we rely on alternative methods to determine information about the qualitative behaviour of solutions to the full equations such as bifurcation theory and stability analysis.

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Section: Discussionmentioning

confidence: 99%

The instability of periodic surface gravity waves

2011

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“…For a fixed µ and a corresponding spectrally stable solution, Ω(ζ) ∈ iR for ζ ∈ R. When ζ ∈ R, the preimage of λ(ζ) ∈ iR is only one point (56) so that by Theorem 1, λ(ζ) is a simple eigenvalue. Therefore, we compute the Krein signature only for ζ ∈ R. From (35),…”

Section: The Advent Of Instabilitymentioning

confidence: 99%

“…When Ω = 0, there exist two bounded eigenfunctions [19,Proposition 3.2]. Only one of these is obtained through (35). We now consider the six values of λ ∈ B.…”

Section: C4 a Proof Of Theoremmentioning

confidence: 99%

The Orbital Stability of Elliptic Solutions of the Focusing Nonlinear Schrödinger Equation

Deconinck¹,

Upsal²

2020

SIAM J. Math. Anal.

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We examine the stability of the elliptic solutions of the focusing nonlinear Schrödinger equation (NLS) with respect to subharmonic perturbations. Using the integrability of NLS, we discuss the spectral stability of the elliptic solutions, establishing that solutions of smaller amplitude are stable with respect to larger classes of perturbations. We show that spectrally stable solutions are orbitally stable by constructing a Lyapunov functional using higher-order conserved quantities of NLS.1. Spectral stability is considered in Section 3. This is motivated by considering the simpler case of the well-known Stokes waves in Section 3.2. For these solutions, all operators involved have constant coefficients, and all calculations are explicit. We get to the spectrum of the operator obtained by linearizing about a solution through its connection with the Lax spectrum. To this end, we introduce the Lax pair and its spectrum in Section 3.3. The results in Section 3.3.1 are from [16] while the results in Section 3.3.2 and all subsequent sections are new. Section 3.4 contains our main spectral stability result: solutions are spectrally stable with respect to subharmonic perturbations if the solution 1

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“…In exactly the same manner, we further obtain new doubly periodic solutions Finally, from (4) and (12), we get nonlinear rational solution u = 8μ 0 x(12μ 0 t + γ 0 ) (μ 0 x 3 + 12μ 0 t + γ 0 ) 2 .…”

Section: Solutions To the Kdv Equationmentioning

confidence: 99%

“…is a typical soliton equation, it's exact expressions of traveling wave solutions have been studied extensively in many papers [1,2,3,4,5,6]. Especially, the linetwo-soliton solutions obtained by the Inverse scattering method in [7] have been well known.…”

Section: Introductionmentioning

confidence: 99%

Exact two waves solutions with variable amplitude to the KdV equation

Huang¹

2014

imf

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By means of the Darboux transformation method, a series of explicit variable amplitude-two-wave solutions to the Korteweg-de Vries (KdV) equation are generated from a particular solution with respect to the independent variable x. Theses new solutions include single soliton solutions, two-soliton solutions, single periodic solutions, doubly periodic solutions. Mathematics Subjection Classification: 35Q51; 35A

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Periodic finite-genus solutions of the KdV equation are orbitally stable

Cited by 23 publications

References 45 publications

The instability of periodic surface gravity waves

The instability of periodic surface gravity waves

The Orbital Stability of Elliptic Solutions of the Focusing Nonlinear Schrödinger Equation

Exact two waves solutions with variable amplitude to the KdV equation

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