Symbolic Computation Of Higher Order Wave Solutions For Higher Dimensional Nonlinear Evolution Equations

Nonlinear evolution equation is a kind of very important mathematical model to describe nonlinear phenomena,and symbolic calculation of exact solutions of nonlinear evolution equations is also an important research topic in the field of mathematical physics.With the rapid development of computer algebra,the computer algebraic systems provide powerful tools and means for people to construct exact solutions of nonlinear evolution equations.In recent years,the research of symbolic computation for high dimensional or super high dimensional nonlinear evolution equations has gradually become a research hotspot in the field of differential equations.Based on the symbolic computation software(Maple),this paper studies symbolic computation of various wave solutions for high-dimensional nonlinear evolution equations,which mainly includes the following two aspects.In the first part,several kinds of higher order wave solutions are constructed by simple Hirota method and direct algebraic method.The simple Hirota method is an effective method to construct exact solutions of the nonlinear evolution equations.However,the formula of N-soliton solution derived from this method usually does not work for non-integrable equations.In this paper,we modify the formula of Nsoliton solution by introducing a kind of parameters constraints to make the modified formula also work well for non-integrable systems.On this basis,combining the Painlevé truncated expansion,long wave limit and conjugated assignment techniques,we investigate arbitrary order solitons,breathers and lumps for the(3+1)-dimensional B-type Kadomtsev-Petviashvili(BKP)equation and the(3+1)-dimensional extended Jimbo-Miwa(JM)equation.Furthermore,based on the direct algebraic method,together with the inheritance solving and parallel computation techniques,we construct higher order rogue wave solutions,interaction solutions between solitons and lump waves for the(3+1)-dimensional BKP equation,and multiwave interaction solutions among solitons,lump waves and periodic waves for the(3+1)-dimensional extended JM equation.In the second part,based on the effective N-soliton solution,we propose an N-soliton decomposition algorithm to construct multiwave interaction solutions of higher order solitons,breathers and lump waves for higher dimensional nonlinear evolution equations.After obtaining the effective multi-soliton solutions,the higher order breathers and lump solutions could be further calculated by the conjugated assignment and long wave limit techniques,respectively.In this paper,a new decomposition algorithm is proposed to construct higher order interaction solutions among breathers,lumps and solitons for multidimensional nonlinear evolution equations.The main idea is to decompose N into: N = 2M+2K+S,where M,K and S are all natural numbers.Then,the former 2M solitons are transformed into M lumps by the long wave limit and conjugated assignment techniques,and the middle 2K solitons are transformed into K breathers by the conjugated assignment technique,and the last S solitons remain unchanged.In this way,we can obtain interaction solutions of M-lumps,K-breathers and S-solitons for multidimensional nonlinear evolution equations.Based on this decomposition algorithm,we construct higher order interaction solutions among lumps,breathers and solitons for the(4+1)-dimensional Fokas equation and the(3+1)-dimensional Generalized KP equation,respectively.