The Traveling Wave Solution And Lie Transformation Method With A Class Of Partial Differential Equations

The main contents of this paper are summarized as follows:The first chapter introduces the development course of soliton theory, mainly solvingmethod and the research scope and application of soliton firstly. Then the development ofLie Qun method and the main idea of exact solution on solving equations arestated.Next development of mathematical mechanization is outlined. Finally a briefdescription on the selection is given.The second chapter takes the Benney equation for example, has used the Tanh functionmethod, the generalized Tanh-function method, the large F expansion method, promotion ofthe large F expansion method and traveling wave solution uniform construction method for itssolving, all these methods are important for traveling wave solution method, and structure issimilar, train of thought come down in one continuous line. By continuing to get new solutionon Benney equation and the similarities and differences of constructive methods, it graduallyreveals the development of constructive skills on traveling wave solution.The third chapter firstly introduces how to use invariance of partial differential equationunder the effect of Lie transformation group to construct its solution. For a given partialdifferential equation, it owns the infinitesimal generator specific methods of Lie transformationgroup, according to the infinitesimal generator, invariant transformation is obtained. Then takefor example specific variable coefficient Vakhnenko equation, by using the above method,invariant transformation is gained. Putting the invariant transformation into the originalequation, which reduces to a ordinary differential equation. Finally, with the help of anBernoulli equation, it constructs special solution of variable coefficient Vakhnenko equations.Symbolic computation software maple is used so multiple times to solve system ofoverdetermined linear partial differential equations during this period. Using abstract algebraLie Qunfa and computer algebra is new development trend of the soliton theory.Finally the work of this paper is summed up, and the future study is proposed...