Generalization Of The Clarkson-Kruskal Direct Method And Applications

Clarkson-Kruskal(CK)direct method is an effective method to dealing with nonlinear evolution equations,which has direct processes and distinct steps,can reduce and construct exact solutions without Lie group theory.Based on symbolic computation software Mathematica and by using the Clarkson-Kruskal direct method,this dissertation investigate the differential-difference(DΔ)equations,the reduction of several types of DΔ equations are presented.Taking the classic Boussinesq equation as an example,the theory and calculation steps of the CK direct method are given.Then we generalize this method to DΔ-Burgers equation,DΔ-Boussinesq equation,DΔ-MKP equation,etc.the corresponding reductions are presented with the following step:in the first step,substitute the intrinsic solution form into the original equation,and get the decision equation with the help of symbolic computation software Mathematica;in the second step,determine appropriate norm coefficients and solve the undetermined functions corresponding to each coefficient according to the four transformation rules;the third step,determine the coefficients,and then obtain the reduction of equations.This dissertation also studies a class of DΔ equations with homogeneous terms,and uses the method of variable separation to solve the first-order Volterra lattice equation.Furthermore,this method is generalized to the second-order BRRT-Toda lattice,coupled nonlinear Toda lattice,Bn-Volterra system and so on.