[Paper Review] On the Solutions of the BBM-KP and BBM model Equations

Jacob Grey, Michael M. Tom|arXiv (Cornell University)|Oct 12, 2014

Nonlinear Waves and Solitons3 references3 citations

TL;DR

This paper establishes the convergence of solutions to the BBM-KP equation toward those of the BBM equation in a suitable function space as the transverse variable $y \to \pm\infty$, provided the initial data for both equations are sufficiently close in this limit. The key contribution is a rigorous asymptotic analysis linking the two models under transverse spatial decay conditions.

ABSTRACT

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 whenever the initial data for both equations are close as the transverse variable $y ightarrow \pm \infty$.

Motivation & Objective

• To investigate the asymptotic relationship between the BBM-KP and BBM equations as the transverse variable tends to infinity.
• To determine under what conditions solutions of the BBM-KP equation approach those of the BBM equation.
• To establish convergence in a suitable function space for the Cauchy problem under transverse spatial decay of initial data.

Proposed method

• Analysis of the Cauchy problem for the BBM-KP equation using functional analytic techniques in a suitable function space.
• Comparison of initial data for the BBM-KP and BBM equations under the condition that they become close as $y \to \pm\infty$.
• Application of energy estimates and asymptotic analysis to control the difference between solutions.
• Use of limiting arguments to show that the BBM-KP solution converges to the BBM solution in the specified function space.

Experimental results

Research questions

• RQ1 Under what conditions does the solution of the BBM-KP equation converge to the solution of the BBM equation as $y \to \pm\infty$?
• RQ2 How does the proximity of initial data at infinity affect the long-term behavior of solutions to the BBM-KP and BBM equations?
• RQ3 What function space framework supports the convergence of solutions between the BBM-KP and BBM models?

Key findings

• Solutions of the BBM-KP equation converge to solutions of the BBM equation in a suitable function space as $y \to \pm\infty$.
• The convergence is established under the condition that the initial data for both equations are close in the limit $y \to \pm\infty$.
• The result holds for the Cauchy problem, indicating global asymptotic stability of the BBM solution under transverse decay of initial data.
• The analysis confirms that the BBM-KP model reduces to the BBM model in the long transverse direction under appropriate initial data conditions.