Application Of The Dbar Method To The Hirota Type Equation

Hirota equation is a kind of high order nonlinear partial differential equation,which has important applications in nonlinear optics and plasma.In this paper,three different types of the Hirota type equations are studied via the Dbar method,including the Hirota equation with zero boundary conditions,the Hirota equation with selfconsistent sources and the Hirota equation with nonzero boundary conditions.Based on a 2×2 matrix Dbar problem,we have obtained the Lax pair by constructing the spectral transformation matrix.According to the compatibility condition of the characteristic function,the suitable spectral transformation matrix is selected to construct the N-soliton solution,and the dynamics behavior of the N-soliton solution is analyzed.Firstly,the classical Hirota equation is studied,and the single soliton solution and two soliton solution are obtained by Dbar method.Then,Hirota equation with self-consistent sources is studied.By considering the odd dispersion relation and adding nonlinear terms,the Hirota equation with self-consistent source is obtained.the N-soliton solution of Hirota equation with self-consistent source is constructed via Dbar method and Cauchy matrix method and the dynamic behavior of the solution is analyzed.It is found that the velocity and trajectory of solitary wave can be changed by self-consistent source.This result has practical significance for controlling solitary wave motion.In addition,the N-soliton solution of Hirota equation in nonzero boundary condition is obtained by using Dbar method.The analytical solutions of equations in nonzero background are more abundant.We find that the different quadrants of discrete spectrum and the choice of nonzero potential affect the interaction between different soliton waves.As the position of a single discrete spectrum changes from the first quadrant to the fourth quadrant,the soliton solutions appear as kink solution-dark soliton solution-kink solution-anti-dark soliton solutions.We also select different parameters to specifically obtain the interaction of kink solution,dark soliton solution and anti-dark soliton solution(strong interaction and weak interaction),and specifically summarize the conditions of pairwise interaction of different type soliton solutions.This study provides a theoretical basis for experiments.