The Asymptotic Approach to the Description of Two-Dimensional Symmetric Soliton Patterns

Yury Stepanyants

doi:10.3390/sym12101586

Abstract

The asymptotic approach is suggested for the description of interacting surface and internal oceanic solitary waves. This approach allows one to describe stationary moving symmetric wave patterns consisting of two plane solitary waves of equal amplitudes moving at an angle to each other. The results obtained within the approximate asymptotic theory are validated by comparison with the exact two-soliton solution of the Kadomtsev–Petviashvili equation (KP2-equation). The suggested approach is equally applicable to a wide class of non-integrable equations too. As an example, the asymptotic theory is applied to the description of wave patterns in the 2D Benjamin–Ono equation describing internal waves in the infinitely deep ocean containing a relatively thin one of the layers.

Highlights

The investigation of solitary waves in oceans has a long story
The asymptotic approach has been developed for the description of stationary patterns formed by two plane solitons interacting at an angle to each other within the KP-type equations
The asymptotic approach has been validated by comparison of the results obtained with the exact two-soliton solution of the completely integrable KP2 equation

Summary

Introduction

The investigation of solitary waves in oceans has a long story. Nowadays, there are several books and reviews justifying that they represent a rather ubiquitous phenomenon on the surface and in the bulk of oceans [1,2]. The theoretical models for the description of surface and internal solitary waves are well-developed, among them the Kadomtsev–Petviashvili (KP) equation; two-dimensional versions of the Gardner, Benjamin–Ono (BO), Joseph–Kubota–Ko–Dobbs (JKKD) equations; and others (see, for example, [1,2] and references therein). In this paper, it KP willmodel, be shown thatthe at existence least for the description symmetric configurations exceptional case of the despite of exact single of soliton solutions within many consisting of two interacting solitary waves, the asymptotic approach can be applied, and the results other models such those mentioned above. The approach is applied to the symmetric configuration of two solitary can be extended to the two-dimensional case too [8] Such an approach has been used for waves interacting at an angle within the 2D version of the BO (2dBO) equation. The approach is applied to the symmetric configuration of two solitary waves interacting at an angle within c ∂ 2η equation

Exact Two-Soliton Solution of the Kadomtsev–Petviashvili Equation

Symmetric

A y

A symmetric Pattern in the 2dBO Equation Formed by Two Plane Solitons

Discussion and Conclusions