A Class Of Nonlinear Partial Differential Equations’ Auxiliary Equation Method And Its Exact Solutions

The world itself is non-linear, and the nonlinear evolution equations as a modelwidely are used to describe the complex physical phenomena. Nonlinear evolutionequations plays a vital role in nonlinear science and engineering. For these models,the basic problem is to get the exact solutions of those nonlinear evolution equations.Soliton solutions of nonlinear sciences, with its high degree of stability and particlecharacteristics,are the favorite of many researchers.All along,in solving nonlinear partial diferential equations, people often getmany soliton solutions, such as single soliton solution,Periodic solution,Two-solitonsolution and the mixed solution. But few people obtained the solutions that containrational function, trigonometric function, hyperbolic function and Jacobi ellipticfunction. So it is our important work to seek various forms of interaction solutionsof nonlinear partial diferential equations.Chapter1,it describes the generation, development and applications of solitontheory, and the soliton theory of seeking exact solutions. Chapter2,it describesthe traditional auxiliary equation expansion method and apply this method to solvethe partial diferential equations with constant coefcients, the partial diferentialequations with variable coefcients, and the discrete partial diferential equations.Chapter3, it describes a new auxiliary equation expansion method, and apply thismethod to obtain new exact solutions of nonlinear partial diferential equations. Chapter4, it describes the two auxiliary equations expansion method and obtain thenew Two-soliton solutions of the (2+1)dimension to Painlev′e Integrable Burgersequation with variable coefcients. Chapter5, it introduces the three auxiliaryequations expansion method, and obtain the new Three-soliton solutions of the (2+1)dimension to Painlev′e Integrable Burgers equation with variable coefcients andthe (2+1) dimension Nizhnik-Novikov-Vesselov equation with variable coefcients.