The Orbit Stability Analysis Of Dark Soliton Of Hirota Equation

As the potential application value of optical soliton technology in the next generation of optical fiber communication becomes increasingly prominent,many mathematicians have begun to work on such nonlinear development equations describing the motion of optical solitons.In this paper,the most representative Hirota equation is used as the research object,focusing on the orbital stability of its dark soliton.The main research contents of the full text are arranged as follows:The first chapter summarizes the development history of solitons、the main research methods and the results obtained,and introduces the methods of constructing conserva-tion laws of nonlinear equations and the research results of the stability of solitary wave solutions.Finally,the main research contents of this paper are briefly described.The second chapter mainly outlines some basic concepts and related preparatory knowledge involved in the study.The third chapter constructs infinite conservation laws based on the following Hirota equation under non-zero background:Where:ψ=ψ(x,t)∈C,(x,t)∈R×R.We First calculate the Lax pairs of the Hirota equation using the AKNS method,then derive the Riccati equation from the x-part of the Lax pairs,and finally construct conservation laws for the Hirota equation using compatibility conditions.The forth chapter focuses on the orbital stability of the dark soliton:ψ=e-it tanh((?))of the following Hirota equation:Assuming that the form of the disturbance at the initial value u0 ψ=u0+u+iv,and then it is proved that u0 is the constrained global minimizer of the higher order conserved quantity of the equation,and finally using the conservation characteristics of higher-order conservation quantity,the orbital stability of the dark soliton in the energy space D under non-zero background is proved.