Validity of Whitham's equations for the modulation of periodic traveling waves in the NLS equation

DOI10.1007/s00332-009-9043-4zbMath1179.35305OpenAlexW2097353563MaRDI QIDQ1041280

Wolf-Patrick Düll, Guido Schneider

Publication date: 2 December 2009

Published in: Journal of Nonlinear Science (Search for Journal in Brave)

Full work available at URL: https://doi.org/10.1007/s00332-009-9043-4

zbMATH Keywords

approximationNLS equationmodulation equationCauchy-Kowalevskaya-like existenceWhitham's equations

Mathematics Subject Classification ID

NLS equations (nonlinear Schrödinger equations) (35Q55) Theoretical approximation in context of PDEs (35A35) Cauchy-Kovalevskaya theorems (35A10)

Related Items

The Validity of Whitham's Approximation for a Klein--Gordon--Boussinesq Model ⋮ Validity of Whitham's modulation equations for dissipative systems with a conservation law: Phase dynamics in a generalized Ginzburg-Landau system ⋮ On the Elliptic-Hyperbolic Transition in Whitham Modulation Theory ⋮ Validity of the hyperbolic Whitham modulation equations in Sobolev spaces ⋮ A proof of validity for multiphase Whitham modulation theory ⋮ Modulational Instability and Variational Structure ⋮ Modulational Instability in the Whitham Equation for Water Waves ⋮ The validity of the KdV approximation in case of resonances arising from periodic media ⋮ Breakdown of the Whitham Modulation Theory and the Emergence of Dispersion ⋮ Validity of the KdV equation for the modulation of periodic traveling waves in the NLS equation ⋮ Dispersive shock waves in lattices: a dimension reduction approach

Cites Work

• Justification of the nonlinear Schrödinger equation in spatially periodic media
• A new estimate for the Ginzburg-Landau approximation on the real axis
• On the validity of the Ginzburg-Landau equation
• Error estimates for the Ginzburg-Landau approximation
• Justification of modulation equations for hyperbolic systems via normal forms
• The rigorous approximation of long-wavelength capillary-gravity waves
• The time dependent amplitude equation for the Swift-Hohenberg problem
• Phase dynamics in the complex Ginzburg-Landau equation
• A proof of the Benjamin-Feir instability
• Macroscopic behavior of microscopic oscillations in harmonic lattices via Wigner-Husimi transforms
• Justification and failure of the nonlinear Schrödinger equation in case of non-trivial quadratic resonances
• The long-wave limit for the water wave problem I. The case of zero surface tension
• ASYMPTOTIC DECAY OF A ONE-DIMENSIONAL WAVE PACKET IN A NONLINEAR DISPERSIVE MEDIUM
• The dynamics of modulated wave trains
• An existence theory for water waves and the boussinesq and korteweg-devries scaling limits
• Semiclassical Soliton Ensembles for the Focusing Nonlinear Schrodinger Equation (AM-154)
• Phase dynamics in the real Ginzburg-Landau equation
• Non-linear dispersive waves
• Micro–macro transition in the atomic chain via Whitham's modulation equation
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item