Convergence of Solutions of the BBM and BBM-KP Model Equations

TL;DR

This paper proves that solutions of the BBM-KP equation converge to solutions of the BBM equation under certain initial conditions, clarifying their relationship in the context of dispersive wave modeling.

Contribution

It establishes the convergence of solutions from the BBM-KP to the BBM equation in a specific function space, enhancing understanding of their mathematical relationship.

Findings

Solutions of BBM-KP converge to BBM solutions as transverse variable tends to infinity.

Convergence holds under initial data closeness conditions.

Provides mathematical foundation for approximating BBM-KP with BBM in certain regimes.

Abstract

The Benjamin-Bona-Mahony (BBM) equation has proven to be a good approximation for the unidirectional propagation of small amplitude long waves in a channel where the crosswise variation can be safely ignored. The Benjamin-Bona-Mahony-Kadomtsev-Petviashvili (BBM-KP) equation is the regularized version of the Kadomtsev-Petviashvili equation which arises in various modeling scenarios corresponding to nonlinear dispersive waves that propagate principally along the $x$-axis with weak dispersive effects undergone in the direction parallel to the $y$-axis and normal to the primary direction of propagation. There is much literature on mathematical studies regarding these well known equations, however the relationship between the solutions of their underlying pure initial value problems is not fully understood. In this work, it is shown that the solution of the Cauchy problem for the BBM-KP equation converges to the solution of the Cauchy problem for the BBM equation in a suitable function space, provided that the initial data for both equations are close as the transverse variable $y \rightarrow \pm \infty$.