On the Korteweg–de Vries Long-Wave Approximation of the Gross–Pitaevskii Equation II

DOI10.1080/03605300903222542zbMath1213.35367arXiv0903.4801OpenAlexW2080839803MaRDI QIDQ3550967

Didier Smets, Philippe Gravejat, Jean Claude Saut, Fabrice Bethuel

Publication date: 8 April 2010

Published in: Communications in Partial Differential Equations (Search for Journal in Brave)

Full work available at URL: https://arxiv.org/abs/0903.4801

zbMATH Keywords

Korteweg-de Vries equationGross-Pitaevskii equationlong-wave regime

Mathematics Subject Classification ID

KdV equations (Korteweg-de Vries equations) (35Q53) NLS equations (nonlinear Schrödinger equations) (35Q55) Theoretical approximation in context of PDEs (35A35)

Related Items

The sine-Gordon regime of the Landau-Lifshitz equation with a strong easy-plane anisotropy ⋮ Asymptotic limit of the Gross-Pitaevskii equation with general initial data ⋮ Recent results for the Landau-Lifshitz equation ⋮ Validity of the Weakly Nonlinear Solution of the Cauchy Problem for the Boussinesq-Type Equation ⋮ From Vlasov-Poisson to Korteweg-de Vries and Zakharov-Kuznetsov ⋮ Logarithmic Gross-Pitaevskii equation ⋮ Long wavelength limit of non-isentropic Euler-Poisson system ⋮ Rarefaction pulses for the nonlinear Schrödinger equation in the transonic limit ⋮ Error bounds for the (KdV)/(KP-I) and (gKdV)/(gKP-I) asymptotic regime for nonlinear Schrödinger type equations ⋮ Stability in the energy space for chains of solitons of the one-dimensional Gross-Pitaevskii equation ⋮ Mean-field dynamics for Ginzburg-Landau vortices with pinning and forcing ⋮ The cubic Schrödinger regime of the Landau-Lifshitz equation with a strong easy-axis anisotropy ⋮ IST Versus PDE: A Comparative Study ⋮ Global Well-Posedness for a Nonlocal Gross–Pitaevskii Equation with Non-Zero Condition at Infinity ⋮ Long wave limit for Schrödinger maps ⋮ On the long-wave approximation for the Euler-Poisson system ⋮ Validity of the KdV equation for the modulation of periodic traveling waves in the NLS equation

Cites Work

• The Cauchy problem for the Gross--Pitaevskii equation
• Orbital stability of traveling waves for the one-dimensional Gross-Pitaevskii equation
• Large time existence for 3D water waves and asymptotics
• Long wave approximations for water waves
• The long-wave limit for the water wave problem I. The case of zero surface tension
• Orbital stability of the black soliton for the Gross-Pitaevskii equation
• An existence theory for water waves and the boussinesq and korteweg-devries scaling limits
• The initial-value problem for the Korteweg-de Vries equation
• Rigorous derivation of Korteweg-de Vries-type systems from a general class of nonlinear hyperbolic systems
• Korteweg-de Vries Equation and Generalizations. II. Existence of Conservation Laws and Constants of Motion
• Corrections to the KdV Approximation for Water Waves