The orbital stability of the cnoidal waves of the Korteweg-de Vries equation

DOI10.1016/j.physleta.2010.08.007zbMath1238.35128OpenAlexW2118173482MaRDI QIDQ418453

Bernard Deconinck, Todd Kapitula

Publication date: 29 May 2012

Published in: Physics Letters. A (Search for Journal in Brave)

Full work available at URL: https://doi.org/10.1016/j.physleta.2010.08.007

Mathematics Subject Classification ID

KdV equations (Korteweg-de Vries equations) (35Q53) Periodic solutions to PDEs (35B10) Water waves, gravity waves; dispersion and scattering, nonlinear interaction (76B15) Nonuniformly hyperbolic systems (Lyapunov exponents, Pesin theory, etc.) (37D25)

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Cites Work

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• The orbital stability of the cnoidal waves of the Korteweg-de Vries equation
• Computing spectra of linear operators using the Floquet-Fourier-Hill method
• Stability of small periodic waves for the nonlinear Schrödinger equation
• Nonlinear stability of periodic traveling wave solutions to the Schrödinger and the modified Korteweg-de Vries equations
• Stability theory of solitary waves in the presence of symmetry. II
• Stability of cnoidal waves
• On the spectra of periodic waves for infinite-dimensional Hamiltonian systems
• Stability of periodic traveling waves for complex modified Korteweg-de Vries equation
• KdV cnoidal waves are spectrally stable
• Stability theory of solitary waves in the presence of symmetry. I
• Computing absolute and essential spectra using continuation
• Orbital stability of periodic waves for the nonlinear Schrödinger equation
• Nonlinear stability of fluid and plasma equilibria
• The Stability Analysis of the Periodic Traveling Wave Solutions of the mKdV Equation
• On the stability ofN-solitons in integrable systems
• The inverse scattering transform: Semi-infinite interval
• Stability for the korteweg-de vries equation
• On the stability of KdV multi‐solitons
• Korteweg-de Vries Equation and Generalizations. IV. The Korteweg-de Vries Equation as a Hamiltonian System
• Sharp global well-posedness for KdV and modified KdV on ℝ and 𝕋
• Elliptic solutions of the defocusing NLS equation are stable
• Integrals of nonlinear equations of evolution and solitary waves
• A presentation of cnoidal wave theory for practical application

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