Numerical solution of the generalized Kadomtsev-Petviashvili equations with compact finite difference schemes

2016/05/09 by Jean‐Paul Chehab, Chehab, J. -P, Pierre Garnier +3
Mathematics · Physics and Astronomy · #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #Differential Equations and Numerical Methods #FOS: Mathematics #Nonlinear Waves and Solitons #Numerical Analysis (math.NA)

paper · pdf · doi:10.48550/arxiv.1605.03213

openalex publication_date 2016/05/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We propose compact finite difference schemes to solve the KP equations u_t + u_xxx + u^p u_x + λ ∂^--1_x u_yy = 0. When p = 1, this equation describes the propagation of small amplitude long waves in shallow water with weak transverse effects. We first present the numerical schemes which are compared to the Fourier spectral method. After establishing the numerical convergence, the scheme is validated. We then depict the behavior of solutions in the context of solitons instabilities and the blow-up.

Related

• Numerical study of the KP equation for non-periodic waves
• A finite difference scheme for smooth solutions of the general mKdV equation
• Numerical study of blow up and stability of solutions of generalized Kadomtsev-Petviashvili equations
• Soliton interactions of the Kadomtsev-Petviashvili equation and generation of large-amplitude water waves
• Discrete conservation laws and the convergence of long time simulations\n of the mKdV equation