The Exact Solution Of Nonlocal Nonlinear Wave Equations And Their Applications

Artificial intelligence in the big data era is constantly evolving,more and more nonlinear effects have been gradually discovered,solving nonlinear equations has always been a hot issue in various areas.Although Darboux transformation and Hirota bilinear method are widely used in solving nonlinear equations,but the Hirota bilinear transformation method and Darboux transformation method are seldom used in solving non-local nonlinear coupled Schr?dinger equation and variable coefficient triple coupled Schr?dinger equation.In view of this kind of problem,this paper studies from the following aspects:In the second chapter,the exact solutions of nonlocal nonlinear Fokas-Lenell equation are solved by using Darboux transformation in the first part.Firstly,starting from a special 2×2 Lax pair,the gauge transformation of the nonlocal nonlinear Fokas-Lenell equation is constructed,and the relation between the new solution and the old solution is derived.The Darboux transformation of the nonlocal nonlinear Fokas-Lenell equation is derived by selecting the zero-seed solution and the non-zero-seed solution respectively.The one-soliton solution,two-soliton solution and N-soliton solution formulas of Fokas-Lenell equation in zero background and non-zero background are obtained by using Darboux transformation.The evolution diagrams of bright soliton,dark soliton,kink soliton and a nti-kink soliton as well as the interaction of two solitons are obtained by using Maple.In the second part,the exact solutions of nonlocal nonlinear MKd V equation are solved by using Darboux transformation,and its one-soliton solution,two-soliton solution and N-soliton solution formula are obtained.The evolution diagrams of periodic wave solution and kink solution are obtained by using Maple,and the dynamic behavior of the inverse space-time solution and the interaction between them are given.The results obtained are different from the classical MKd V equation.In the third chapter,the nonlocal nonlinear coupled Schr?dinger(RS-NCNLS)equation with3×3 spectral problem is further considered.According to the given 3×3 Lax pair,the Darboux transformation of the nonlocal nonlinear coupled Schr?dinger equation is constructed.then the formulas of one-soliton solution,two-soliton solution and N-soliton solution of the nonlocal nonlinear coupled Schr?dinger equation under zero background and non-zero background are obtained by Darboux transformation.Using Maple,the interactions between these soliton solutions are shown by selecting the same and different parameters.In the fourth chapter,the bilinear method is used to solve the combined Kd V-MKd V equation in the first part.Firstly,the combined Kd V-MKd V equation is transformed into a bilinear equation by using bilinear operators.Then the solution of the equation is set as the combination form of hyperbolic function and trigonometric function,the combination form of exponential function and periodic function,and the one-soliton solution of the Kd V-MKd V equation is obtained.The second part studies the variable coefficient triple coupling Schr?dinger equation.Firstly,the relationship between the variable coefficient triple coupling Schr?dinger equation and the constant coefficient triple coupling Schr?dinger equation is given by using the similarity transformation,then the bilinear method is used to get the solution of the constant coefficient equation.Finally,the expression of the solution of the variable coefficient equation is given.The parabolic and periodic solutions,as well as the collision between solitons are shown in Maple.