Symmetry, Exact Solutions And Application For Several Nonlinear Partial Diferential Equations

In the development of mathematical physics, it is well known that many nonlinear partialdiferential equations are widely used as models to describe complex physical phenomena invarious fields of sciences, especially in fluid mechanics, solid state physics, plasma physics, biol-ogy and so on. Searching for exact solutions of nonlinear partial diferential equations by usingvarious methods has become an extremely valuable task, and many powerful and efcient meth-ods methods to construct exact solutions of nonlinear partial diferential equations have beenestablished and developed by a diverse group of scientists, such as the inverse scattering trans-form, the Darboux transformation, B¨acklund transformation method, Hirota bilinear method,variable separation method, the tanh function method, symmetry group method, Painlev′e ex-pansion method,sub-equation expansion method, Exp-function method and so on. In recentyears, due to the availability of symbolic computation systems like Mathematica or Maple withthe development of computer science, direct methods to search for exact solutions of nonlinearpartial diferential equations has attracted more and more attention. In this paper, we em-ploy extended multiple (G′/G)-expansion method, generalized sub-equation method, symmetrygroup, Modified CK direct method, Painlev′e expansion method, Hirota bilinear method andRiemann theta function to investigate several nonlinear partial diferential equations.The thesis is arranged as follows:Chapter1Make a berif introduction about soliton and several methods for searching ana-lytical solutions of nonlinear partial diferential equations in this chapter.Chapter2An extended multiple (G′/G)-expansion method is proposed to seek exact so-lutions of nonlinear evolution equations. The validity and advantages of the proposed methodis illustrated by the applications to the (1+1)-dimensional nonlinear evolution equations: theSharma-Tasso-Olver equation, the sixth-order Ramani equation, the generalized shallow waterwave equation, the Caudrey-Dodd-Gibbon-Sawada-Kotera equation, the sixth-order Boussinesqequation and the Hirota-Satsuma equations;the (2+1)-dimensional nonlinear evolution equa-tions:the integrable generalized Sawada-Kotera equation, the Painlev′e integrable Burgers equa-tion, the Asymmetric Nizhnik-Novikov-Vesselov equation, the Breaking Soliton equation and theNizhnik-Novikov-Veselov equation; the (3+1)-dimensional nonlinear evolution equations:Jimbo-Miwa equation and the (3+1)-dimensional Burgers equation. As a result, various of complexitonsolutions consist of hyperbolic functions, trigonometric functions, rational functions and theirmixture with parameters are obtained. When some parameters are taken as special values, the known double solitary-like wave solution are derived from the double hyperbolic functionsolution.Chapter3Based on the generalized symmetry group method, the finite symmetry trans-formation groups of the Wu-Zhang equation and (2+1)-dimensional Caudrey-Dodd-Gibbon-Sawada-Kotera equation are presented by symbolic computation. Some complexiton solutionsof Wu-Zhang equation are derived by the multiple Riccati equations rational expansion method.The Caudrey-Dodd-Gibbon-Sawada-Kotera equation can be proved to be Painlev′e integrabilityby combining the standard WTC approach with the Kruskal’s simplification, some solutions areobtained by using the standard truncated Painlev′e expansion.Chapter4The generalized sub-equation method is extended to investigate localized nonlin-ear wave of the one-dimensional nonlinear Schro¨dinger equation with potentials and nonlineari-ties depending on both time and spatial coordinates. Based on three families of analytical solu-tions presented by symbolic computation, periodically and quasiperiodically oscillating solitons(dark and bright) and moving solitons are observed. A similarity transformation reducing the(3+1)-dimensional inhomogeneous coupled nonlinear Schr¨odinger equation with variable coef-cients and parabolic potential to the (1+1)-dimensional coupled nonlinear Schr¨odinger equationwith constant coefcients is systematically provided. The dynamics of the propagation of thethree-dimensional bright-dark soliton, the interaction between two bright solitons and the fea-ture of the three-dimensional rogue wave with diferent parameters are discussed.We investigatethe two-dimensional spatially inhomogeneous cubic-quintic nonlinear Schr¨odigner equation withdiferent external potentials. Without external potential or in the presence of harmonic poten-tial, the number of localized nonlinear waves is associated not only with the boundary conditionbut also with the singularity of inhomogeneous cubic-quintic nonlinearities; while in the presenceof periodic external potential, the periodic inhomogeneous cubic-quintic nonlinearities, togetherwith the boundary condition, support the periodic solutions with arbitrary number of circularring in every unit.Chapter5By Hirota bilinear method and Riemann Theta function, multi-periodic wavesolutions are constructed for the (1+2)-dimensional Ito equation. A limiting procedure is pre-sented to analyze in detail, in which the asymptotic behavior of multi-periodic waves and therelations between the periodic wave solutions and soliton solutions are rigorously established.