Decoupled and unidirectional asymptotic models for the propagation of internal waves

Abstract

We study the relevance of various scalar equations, such as inviscid Burgers', Korteweg-de Vries (KdV), extended KdV, and higher order equations (of Camassa-Holm type), as asymptotic models for the propagation of internal waves in a two-fluid system. These scalar evolution equations may be justified with two approaches. The first method consists in approximating the flow with two decoupled, counterpropagating waves, each one satisfying such an equation. One also recovers homologous equations when focusing on a given direction of propagation, and seeking unidirectional approximate solutions. This second justification is more restrictive as for the admissible initial data, but yields greater accuracy. Additionally, we present several new coupled asymptotic models: a Green-Naghdi type model, its simplified version in the so-called Camassa-Holm regime, and a weakly decoupled model. All of the models are rigorously justified in the sense of consistency.

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• On the persistence properties of solutions of nonlinear dispersive equations in weighted Sobolev spaces
• Well-posedness and stability results for the Gardner equation
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Cited by in corpus (8)

• A new class of two-layer Green-Naghdi systems with improved frequency dispersion
• A new fully justified asymptotic model for the propagation of internal waves in the Camassa-Holm regime
• Shallow water asymptotic models for the propagation of internal waves
• On the rigid-lid approximation for two shallow layers of immiscible fluids with small density contrast
• Higher-order Hamiltonian Model for Unidirectional Water Waves
• Medium amplitude model for internal waves over large topography variation
• On the decoupling of the improved Boussinesq equation into two uncoupled Camassa-Holm equations
• On sharp global well-posedness and Ill-posedness for a fifth-order KdV-BBM type equation