Some Exact Solutions For Complex Nonlinear Soliton Equations

Nonlinear waves are ubiquitous in physical systems,and nonlinear equations also play an important role in various fields of physics.It is of great significance to explore the exact solution of nonlinear equations.In the current study of soliton theory,more and more scholars are devoted to solving the exact solution of nonlinear equations.However,there are few studies on solving the nonlocal nonlinear Schr?dinger equation and coupled nonlocal variable coefficients equation by using the Hirota bilinear method and Darboux transformation(DT).In view of the above problems,this paper will carry out the following research.In the second chapter,we first solve the non-autonomous wave solutions of the Gross-Pitaevskii(GP)equation with parabola external potential by using the homogeneous balance method and the F-extension method.On the basis of the similarity transformation,the non-autonomous wave solutions of GP equations with snaking behaviors and different amplitudes are given,and the bright and dark soliton solutions are derived,they have some potential applications in Bose-Einstein condensates.A new mixed local-nonlocal Schr?dinger(MLN-NLS)equation is proposed.We investigate the Hirota’s bilinearization transformation and obtain the one-and two-soliton solutions of the MLN-NLS equation.Based on the solutions,the propagation and interaction structures of these solitons are shown,the broken and unbroken soliton solutions are derived by Hirota’s transformation.We find that unlike the local or nonlocal cases,the MLN-NLS equation has some novel results.Finally,the coupled variable coefficients nonlinear Schr?dinger equation is studied.Then one-,two-and N-soliton solutions are obtained.By constructing the bilinear form,the respiration solution and soliton solution are derived,and the graphs are shown by the Maple.These results enrich the understanding of the dynamics of soliton propagation in physics.In the third chapter,the exact solutions of nonlinear Kundu-Eckhaus(KE)equation are solved by using Darboux transformation.Firstly,the equation is derived from the zero curvature equation.Then we find a special Lax pair to construct the canonical transformation T,and obtain the recursion formula of the matrix T after substituting the initial value.The relation between the new solution and the old solution is found,and the Darboux transformation of KE equation is derived by using the iterative technique and the determinant.Then the one-,two-and N-soliton solutions are obtained.The interactions between solitons are shown.Secondly,the Darboux transformation and the exact solutions of the three coupled Schr?dinger equations for the 4×4 matrix spectral problem are investigated.Based on the transformation between Lax pairs,the Darbourx transformation of the three coupled Schr?dinger equations are constructed,and the one-and two-soliton solutions are obtained.