Nonlocal Symmetry And Exact Solutions Of Nonlinear Differential Systems

Nonlinear differential systems are widely used in many scientific and engineering fields.Studying the exact solutions of nonlinear systems can further explore the properties and physical significance of equations.In this paper,with the help of symbolic computing software Maple,the nonlocal symmetry,interaction solution,multi soliton solution,lump solution and trajectory equation of a lump solution before and after collision with other nonlinear waves for nonlinear partial differential equations are studied.The details are as follows:Firstly,the residual symmetry of the(1+1)-dimensional dispersive long-wave system are researched by using the truncated Painlevé expansion method,and it is proved that this system is consistent Riccati expansion(CRE)solvable.Then,through the special simplified form of CRE-consistent tanh expansion(CTE)solvability,the soliton-cnoidal interaction solutions of(1+1)-dimensional dispersive long-wave system are obtained,and the corresponding image simulations are provided by selecting appropriate parameters.Secondly,based on the truncated Painlevé expansion method,the residual symmetry of the(2+1)-dimensional Kadomtsev-Petviashvili(KP)system is derived.With the help of Lie’s first fundamental theorem,the finite symmetry transformation of the KP system is presented.Then,the CRE solvability of this system is proved,and the interaction solutions between the soliton and cnoidal waves in the(2+1)-dimensional KP system are obtained by using the compatibility equation.Finally,by using the Hirota bilinear and long wave limit methods,the hybrid solutions of M-order lumps with n solitons and n breather waves for generalized(2+1)-dimensional Hirota-Satsuma-Ito(GHSI)equation are constructed.By approximating solutions of the GHSI equation along some parallel lines at infinity,the trajectory equation of a lump wave before and after collision with n solitons and n breather waves are studied,and the expression of phase shift for the lump wave before and after collision are given.At the same time,it is revealed that collisions between the lump wave and other waves are elastic,and the corresponding collision diagrams are used to explain.