Exact Solutions Of Nonlinear Wave Equations And Their Soliton Structures

In nonlinear science,the study on the exact solution of nonlinear wave equations is helpful in clarifying the underlying algebraic structure of the soliton theory and plays an important role in reasonable explaining of the corresponding natural phenomenon and application.In this dissertation,with the help of the symmetry reduction theory in nonlinear physics and the variable separation approach in linear physics,the direct albebra method and the multilinear variable separation approach are studied and extended to nonlinear physics successfully, then a new algorithm - the mapping transformation approach is proposed and many kinds of new results are obtained from there.To seekig for the exact travelling wave solution and approximate solution,the algebra method based on travelling wave reduction is extended and applied to some nonlinear discrete systems and complicated equations.Based on the mapping transformation solutions and the multilinear variable separation solutions respectively, abundant localized excitations and related nonlinear dynamical behaviors for some (2+1)-dimensional(two spatial-dimensions and one time dimension) nonlinear models are investigated. The localized excitations and related fractal and chaotic behaviors in nonlinear wave equations are discussed,which are orginated from many natural sciences,such as fluid dynamics,plasma physics,solid physics,superconducting physics,condensed matter physics and optical problems.The research results indicate that one can establish the relationship among the Charkson-Kruskal(CK) direct reduction method,the mapping transformation approach and the multilinear variable separation approach.Meanwhile,fractals and chaos in higher-dimensional nonlinear soliton systems are quite universal phenomena.The main contents are summarized as folows:In the first chapter,a brief history of finding solitary waves and solitons is outlined and several improtant methods for studying soliton solutions are listed,including the inverse scatterng method,the Darboux transformation and the Backlund transformation,the Painlevéanalysis approach and the Hirota bilinear method.The research arrangement of the dissertation is given in the end of this chapter.In the second chapter,taking an example of the Boussinesq equation,three powerful methods are presented for finding similarity reduction solutions of a nonlinear wave equa- tion,i.e.,the Lie group method of infinitesimal transformations,the nonclassical Lie group method and the Clarkson and Krnskal direct method.Starting from the Lax expression of the(2+l)-dimensional sine-Gordon system,a modified technic for the CK direct method was put forward by Lou and Ma recently.The symmetry group and then the Lie symmetries and the related algebra can be reobtained via a simple combination of a gauge transformation of the spectral function and the transformations of the space time variables.Meanwhile,applying the group transformation theorem on the multiple straight line soliton solutions,one can obtain various types of multiple curved line excitations.Finally,the objective reduction approach is presented based on the idea of the above CK direct method.The main theory of this method is:For a given nonlinear equation,an objective function is first established, supposing that the obtained similarity reduction equation is an ordinary partial equation, through setting one coefficient as the normalizing coefficient in each part of this equation,we may separate all coefficients into several parts and require that the ratios of these coefficients satisfy the same objective reduction equation.From there,the abundant exact solutions of a given system are derived,including solitary wave solutions,periodic wave solutions and rational function solutions.In the third chapter,a new mapping transformation for a nonlinear equation is proposed using the CK direct similarity reduction theory.The approach breaks through the traditional idea,which only the travelling wave solution of a nonlinear equation can be obtained,and is applied to some nonlinear wave equations,such as the(2+1)-dimensional dispersive long wave system,the(2+1)-dimensional generalized Broer-Kaup system,the(1+1)-dimensional nonlinear Schrodinger system,the(2+1)-dimensional modified dispersive water-wave system, and the(2+1)-dimensional Nizhnik-Novikov-Veselov system.Based on the new variable separation solutions,some new or typical localized coherent excitations and their evolution properties are revealed.By introduing suitable arbitrary functions,considerably novel localized structures are constructed,such as doubly periodic patterns from the Jacobi elliptic functions,semifolded localized structures including multi-valued and single-valued solitons, and certain localized excitations with fission and fusion behaviors.Some typical localized excitations with fractal property and chaotic behavior are also discussed.Why the localized excitations possess such kinds of chaotic behavior and fractal property? If one considers the boundary or initial condition of the chaotic and fractal solutions obtained here,one can find straightforwardly that the initial or boundary condition possesses chaotic and fractal properties.These chaotic and fractal properties of the localized excitations for an integrable model essentially come from certain "nonintegrable" chaotic and fractal boundary or initial condition.From these theoretical results,one may interpret that chaos and fractals in higherdimensional integrable physical models would be a quite universal phenomenon.Therefore, all the localized excitations based on the multilinear variable separation solution can be rederived by the mapping transformation solution.Meanwhile,we have established a simple relation between the multilinear variable separation solutions and the mapping transformation solutions,which are essentially equivalent by taking certain variable transformation.The mapping transformation approach not only outbreaks its original limitation merely searching for traveling wave solutions to nonlinear systems,but also can be extended to many other nonlinear dynamical systems,which also means that the mapping approach has been richened and developed to the basic theory of nonlinearity.In the fouth chapter,several studying aspects of the variable separation method are first introduced briefly,and then the general process for a(2+1)-dimensional nonlinear system of the Lou's multilinear variable separation approach is given.As a result,the variable separation solutions and their generalizations of the(1+1)-dimensional shallow water wave equation, the(2+1)-dimensional mBK equations and some other famous equations are solved.Some localized excitations and their interaction behaviours are revealed under a quite "universal" variable separation formula by selecting appropriate initial and/or boundary conditions. Based on the plots and theoretical analysis,we explored some typical localized excitations. Dromions are localized solutions decaying exponentially in all directions,which can be driven not only by straight line solitons but also driven by curved line solitons and can be located not only at the cross points of the lines but also at the closed points of the curves.Foldons are a class of multi-valued solitary waves,which can be folded in all directions.The fractal solitons and chaotic solitons reveal fractal characteristics and chaotic dynamic behaviors in solitary waves,respectively.The dipole-type dromion and the paraboloid-type camber soliton solution are plotted in their equipotential form or projective density form for the(3+1)-dimensional Burgers equation.In the fifth chapter,to seekig for the exact travelling wave solution and approximate solution for some nonlinear discrete systems and complicated equations,the algebra method based on travelling wave reduction is extended and applied to them.The tanh function approach is devised for the nonlinear differential-difference equations,its two generalized applications are presented.The deformation mapping theory based on travelling wave reduction and the reduction solutions of the nonlinear Schrodinger equation are given.At last,the Adomian decomposition method is implemented for solving a hlgher-order nonlinear Schrodinger equation in atmospheric dynamics with the initial condition to obtain the approximate solutions based on the travelling wave transformation.Finally,some main and important results as well as future research topics are outlined in the last chapter.