The Breather-Like and Rational Solutions for the Integrable Kadomtsev-Petviashvili-Based System

Chuanjian Wang,Changfu Liu,Zhengde Dai

doi:10.1155/2015/861069

Abstract

The integrable Kadomtsev-Petviashvili-based system is studied. The breather-like (a pulsating mode) and rational solutions are presented applying Hirota bilinear method and Taylor series. The intricate structures of the rational solitary wave solution are discussed mathematically and graphically. The existence conditions of three different solitary wave solution structure for the short-wave field are given by the theory of extreme value analysis. By controlling the wave number of the background plane wave we may control the the behavior of rational solitary wave. However, the shape of the rational solitary wave solution for the real long-wave field is not affected as the wave number is varied.

Highlights

IntroductionThe integrability property and Lax pairs of system (1) were obtained in [1]
The integrable Kadomtsev-Petviashvili-based system is derived from the Kadomtsev-Petviashvili (KP) equation via an asymptotically exact reduction method based on the Fourier expansion and spatiotemporal rescaling [1]; it can be written in the following form: iut + uxx + uV = 0, (1) Vt + Vy (|u|2 ) x =0, where i = √−1, u(x, y, t) is a complex function of two scaled space coordinates x, y and time t, and V(x, y, t) is a real one
(iii) If δ2 ∈ [3, +∞), the function u(x, y, t) shows dark solitary wave feature (see Figure 1(c)). These analyses show that the structures of the breatherlike and rational solutions are instable

Summary

Introduction

The integrability property and Lax pairs of system (1) were obtained in [1] It is a family of nonlinear partial equation that are often used to describe the phenomenon in the relevant physical fields such as nonlinear optic, plasma physics, and hydrodynamic [2]. It has been studied by many authors. Employing the Taylor series or the long wave limits [7,8,9,10,11], the rational solution is obtained This kind of solution is called rogue wave [8,9,10,11]. Advances in Mathematical Physics evolution equations exhibit rogue wave solutions, for example, the focusing and defocusing Ablowitz-Ladik equations [14], Davey-Stewartson equation [15], coupled SchrodingerBoussinesq equation [16], Sasa-Satsuma equation [17], coupled nonlinear Schrodinger and Maxwell-Bloch equations [18], systems displaying PT-symmetry [19], three-component coupled nonlinear Schrodinger equation [20], and even modeling in finance [21]

Breather-Like and Rational Solutions

Rational Solution of the Standard KP Equation

Conclusions