Approximation of the Korteweg-de Vries equation by the nonlinear Schrödinger equation

DOI10.1006/jdeq.1998.3417zbMath0940.35179OpenAlexW2047012233MaRDI QIDQ1268392

Guido Schneider

Publication date: 19 July 2000

Published in: Journal of Differential Equations (Search for Journal in Brave)

Full work available at URL: https://doi.org/10.1006/jdeq.1998.3417

zbMATH Keywords

approximationKdV equationNLS equationwater wave problem

Mathematics Subject Classification ID

KdV equations (Korteweg-de Vries equations) (35Q53) Water waves, gravity waves; dispersion and scattering, nonlinear interaction (76B15) NLS equations (nonlinear Schrödinger equations) (35Q55)

Related Items

Justification of the nonlinear Schrödinger equation for the evolution of gravity driven 2D surface water waves in a canal of finite depth ⋮ Conservation of resonant periodic solutions for the one-dimensional nonlinear Schrödinger equation ⋮ Validity of the resonant four-wave interaction system in a model for surface water waves on an infinite deep sea ⋮ An Embedded Exponential-Type Low-Regularity Integrator for mKdV Equation ⋮ Justification of the nonlinear Schrödinger approximation for a quasilinear Klein-Gordon equation ⋮ Existence of a minimal non-scattering solution to the mass-subcritical generalized Korteweg-de Vries equation ⋮ Gauge-Transformed Exponential Integrator for Generalized KdV Equations with Rough Data ⋮ Validity of the nonlinear Schrödinger approximation for the two-dimensional water wave problem with and without surface tension in the arc length formulation ⋮ Note on nonlinear Schrödinger equation, KdV equation and 2D topological Yang–Mills–Higgs theory ⋮ Numerical study of oscillatory regimes in the Kadomtsev-Petviashvili equation ⋮ Justification of the NLS approximation for the KdV equation using the Miura transformation ⋮ Justification of the NLS approximation for a quasilinear water wave model ⋮ Justification of the NLS approximation for the Euler-Poisson equation ⋮ Justification and failure of the nonlinear Schrödinger equation in case of non-trivial quadratic resonances

Cites Work

• Unnamed Item
• Unnamed Item
• Unnamed Item
• Multi-scale expansions in the theory of systems integrable by the inverse scattering transform
• On the validity of the Ginzburg-Landau equation
• Error estimates for the Ginzburg-Landau approximation
• The time dependent amplitude equation for the Swift-Hohenberg problem
• An existence theory for water waves and the boussinesq and korteweg-devries scaling limits
• Nonlinear modulation of gravity waves: a rigorous approach
• The validity of modulation equations for extended systems with cubic nonlinearities
• Long wave asymptotics. Integrable equations as asymptotic limits of non-linear systems
• Validity and Limitation of the Newell‐Whitehead Equation
• Finite bandwidth, finite amplitude convection