Study On Some Problems Of Nonlocal Symmetry And Nonlinear Waves

Based on symbolic computation,this dissertation studies the symmetry,nonlinear waves and wave breaking phenomena of some nonlinear integrable models by using nonlocal symmetry,bilinear and characteristic methods.It mainly includes five aspects: interaction solutions and localized excitations of the integrable systems are studied by using nonlocal symmetry method;consistent Riccati expansion(CRE)method is used to investigate the solvability and exact solutions of the systems;solitary wave,lump,breather and rogue wave,four types of nonlinear waves are derived by virtue of Hirota bilinear method;two Maple packages named LumpSol and InterSol are developed to construct lump solution and interaction solution between the lump and soliton for the integrable equations,respectively;wave breaking phenomenon and the existence condition of global strong solution for a Camassa-Holm type shallow water model with coriolis effect are studied based on the method of characteristics.The details are as follows:Chapter 1 is an brief introduction of the background and development status of symmetry theory,nonlinear wave,shallow water wave with coriolis effect and symbolic computation.The main research results of this dissertation are also elaborated.In chapter 2,nonlocal symmetries,prolonged systems,similarity reductions and interaction solutions for the Drinfeld-Sokolov-Satsuma-Hirota(DSSH)system,2+1-dimensional KdV equation and the reduced Maxwell-Bloch(RMB)equations are investigated,localized excitations of the RMB equations are studied by the nonlocal symmetry method for the first time.Firstly,nonlocal symmetries of the DSSH system are constructed by the Lax pair;nonlocal symmetries of the 2+1-dimensional KdV equation are given by the Lax pair and truncated Painlev?e expansion methods;nonlocal symmetries of RMB system are obtained by the truncated Painlev?e expansion method.By the two approaches,the nonlocal symmetries and reduced Schwarz form of the original system are consistent.Here rich exact interaction solutions are derived between solitary waves and other waves including cnoidal waves,rational waves,Painlev?e waves,and periodic waves through similarity reductions.Specially,several localized excitations including rogue waves,breathers and other nonlinear waves for the RMB equations are obtained.In chapter 3,CRE solvability and exact solutions of the generalized KadomtsevPetviashvili(gKP)equation,modified Bogoyavlenskii-Schiff(mBS)equation and RMB equations are studied.Firstly,consistent Riccati expansions of the three equations are constructed by the truncated Painlev?e expansion method.CRE solvability and consistent tanh expansion(CTE)solvability are verified.Then,solitary waves and interaction solutions between solitary waves and cnoidal waves are obtained applying CTE method.Finally,graphics are presented to demonstrate the dynamic behaviors and properties of interaction solutions.In chapter 4,nonlinear waves of the 3+1-dimensional gKP equation and 2+1-dimensional Sawada-Kotera equation are studied.Firstly,based on the Hirota bilinear and long wave limit methods,four kinds of nonlinear waves,namely,solitons,lumps,breathers,rogue waves and interaction solutions are constructed for the 3+1-dimensional gKP equation.Then lump and interaction solutions between lump and line soliton of the 2+1-dimensional Sawada-Kotera equation are derived by Hirota bilinear and function quasi methods,the interaction solution is found to be a completely inelastic collision.Finally,two Maple packages named LumpSol and InterSol are developed to construct lump and interaction solution between lump and soliton for integrable systems,respectively.The validity and convenience of the two software packages are verified by some examples.In chapter 5,a Camassa-Holm type shallow water model with Coriolis effect is studied,which can be derived as an asymptotic model for the propagation of long-crested shallow-water waves in the equatorial ocean regions with the weak Coriolis effect due to the Earth's rotation,and is also related to the compressible hyperelastic rod model in the material science.This model has a formal Hamiltonian structure and its solution corresponding to physically relevant initial perturbations is more accurate on a much longer time scale.It is shown that the solutions blow up in finite time in the sense of wave breaking.A refined analysis based on the local structure of the dynamics is performed to provide the wave breaking phenomena.The effects of the Coriolis force caused by the Earth's rotation and nonlocal higher nonlinearities on blow-up criteria and wave breaking phenomena are also investigated.Finally a sufficient condition for global strong solutions to the equation in some special cases is given.In chapter 6,the summary and discussion of this dissertation are given,and the outlook of future work is also discussed.