Application Of The Multilinear Variable Separation Approach In (3+1)-Dimensional Nonlinear Systems

In order to better understand various complex nonlinear physical phenomena in nature,nonlinear systems enter the scientists' view.Looking for solutions of nonlinear systems is an important topic in nonlinear science.People have established many methods for solving nonlinear systems.Multi-linear variable separation approach(MLVSA)is one of the most effective methods,playing a great role in solving low-dimensional systems.However,there are fewer applications in high-dimensional situations.By means of the symbolic computing software platforms Maple and Mathematica,this dissertation mainly studies the application of MLVSA in(3+1)-dimensional nonlinear systems.Firstly,use the MLVSA to solve the(3+1)-dimensional Boiti-Leon-Manna-Pempinelli(BLMP)equation and a(3+1)-dimensional nonlinear evolution equation,and multi-linear separated variable solutions with arbitrary background waves are obtained.Secondly,several special representations of the arbitrary functions in the multi-linear separation variable solution are given to constructe N × M lattice solitary waves with the dromions,lumps and ring solitons excited on the lattices.The parameters are adjusted to draw lattice solitary waves,and their interaction rule is thus obtained.Recently,soliton molecules have attracted people's great attention.A soliton molecule is resonance state of solitons,and the velocity resonance is an important mechanism for binding two or more solitons to form a soliton molecule.By requiring the arbitrary parameters of the solitons in the lattice solitary waves satisfy the velocity resonance condition,abundant soliton molecules are constructed,including the symmetric and asymmetric dromion soliton molecules,symmetric and asymmetric lump soliton molecules,symmetric and asymmetric ring solitons molecules.The computer software platform Mathematica is used to draw their three-dimensional images.The dynamic behavior and characteristics of the interactions between soliton and soliton molecules are discussed.Finally,the process of solving the(3+1)-dimensional nonlinear system with MLVSA is algorithmically programmed.As an illustration,a program package named MULTILINEAR is completed for the BLMP equation,and how to invoke the MULTILINEAR program package is discussed.Through the function call,different parameter matrices can be input to draw the three-dimensional structures and their interactions for a variety of nonlinear localized excitations.The programme provides convenience for studying dynamic behaviors of localized waves.