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

TL;DR

This paper develops compact finite difference schemes for solving the generalized Kadomtsev-Petviashvili equations, compares them with spectral methods, and analyzes solution behaviors like soliton stability and blow-up phenomena.

Contribution

It introduces new compact finite difference schemes for KP equations, demonstrating their convergence, validation, and effectiveness compared to spectral methods.

Findings

Numerical schemes are validated through convergence analysis.

Schemes effectively capture soliton behaviors and instabilities.

The methods provide accurate solutions for KP equations with different parameters.

Abstract

We propose compact finite difference schemes to solve the KP equations $u\_t + u\_{xxx} + u^p u\_x + $\lambda$ \partial^{--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.