BT Of Variable-Coefficient KdV Hierarchy And Exact Solutions Of Heat-Like And Wave-Like Equation

With the deep study of soliton theory,it is found that there is soliton solutions in completely integrable equations in nonlinear systems.In recent several decades,there have been many methods for solving exact solutions of soliton equations,such as inverse scattering method,Hirota bilinear method,B?cklund transformation(BT)and so on.Among them,the inverse scattering method and Hirota bilinear method play indelible roles in the development of soliton theory.It is found that the inverse scattering method can also be used to solve the equation of BT.Hirota bilinear method belongs to one of constructive methods.The advantage of this method is that it does not depend on the Lax pair or spectral problem associated to the equation,this make the method simple and intuitive.Compared with the bilinear method,the superiority of the BT is that another solution can be written directly by the obtained solution and the relationship between them,and then many new solutions are obtained.But the difficulty is that it is difficult to find the transformation for the relationship.Scholars have discovered the bilinear form of BT by studying,which makes the solution simpler and more explicit.In recent years,the fractional problems have aroused more and more attention.Exactly solving the fractional differential equations is difficult and is still in the initial stage.The main work of this dissertation is summarized as follows:Firstly,from the Schr?dinger linear spectrum problem and its time development,i.e.Lax pair,by using the compatibility conditions of eigenfunctions,a new Lax integrable variable coefficient KdV hierarchy is introduced by introducing a finite number of only time dependent functions.At the same time,we obtain the bilinear form BT of a variable coefficient KdV equation in the variable coefficient KdV hierarchy.Based on the BT found,we obtain the soliton solution of the variable coefficient KdV equation,the double soliton solution,the three soliton solution,and then sum the n soliton solution.Secondly,a classs of variable-coefficient Cauputo's time-fractional heat-like and wave-like equation with initial and boundary conditions is solved by the use of variable separation method and the properties of Mittag–Leffler function.As a result,a uniform expression of exact solution of the class of variable-coefficient time-fractional heat-like and wave-like equation are obtained.In order to further determine the exact solution,three specific heat-wave equations with initial boundary conditions are considered,and their exact solutions are obtained,from which some known special solutions are recovered.It is shown that the variable separation method can also be used to solve some others time-fractional heat-like and wave-like equations in science and engineering.