Interaction Solutions For Nonlinear Integrable Systems

Nonlinear evolution equations can be used to describe a large number of nonlinear phenomena in nonlinear science.Soliton theory is an important branch of nonlinear science,its main content is the exact solution of nonlinear evolution equation and its integrability analysis.However,due to the physical background of nonlinear evolution equation and the complexity of its solution,there is no unified method to solve it up to now.In this thesis,a constructive method for two classes of nonlinear integrable model problems is established by using the characteristic of large capacity and high speed of computer and the accurate symbolic calculation software Maple,and the interaction solutions of these two classes of nonlinear integrable systems are studied by using this method.The main research contents are as follows:In the first part,based on the integrability of the equations and the relationship between the equation Lax pair and the nonlinear integrable system,Two algorithm are designed on Maple to automatically derive the Lax pair of the equations and verify the correctness of the Lax pair.These algorithms are also used to study the Sawada-Kotera(SK)equation and Kaup-Kupershmidt(KK)equation of 1+1 dimension 5th-order.The parameters of the equation in the form of L and M are determined by the Laxpaircompute package to obtain the Lax pairs of the equation,and verify the correctness of the Lax pairs by the Laxpairtest package.In the second part,the bilinear method is used to set the untested function f as the interaction function for the periodical wave and the solitary wave.This function is further used to study the solution of periodic solitary wave for the 1+1 dimension 5th-order Sawada-Kotera(SK)equation and 2+1 dimension nonlinear wave equation through selecting different parameters of the periodic solitary wave solution,the corresponding evolution diagram is obtained.With the change of the parameters,it is obvious that the wave evolution changes from one kind of process to another,the inherent law of nonlinear wave is revealed.In the third part,the bilinear method is used to set the untested function f as the interaction function for quadratic function and exponential function.This function is further used to study the lump-strip solutions of 1+1 dimension 5thorder Sawada-Kotera(SK)equation and 2+1 dimension nonlinear wave equation,the evolution diagram is obtained by selecting the values of different parameters in the lump-strip,showing the process of engulfment and fusion in the collision of lump and solitary waves,which is a kind of Inelastic collision,by changing the values of the parameters,the bright lump solution of the equation is also obtained.