Soliton Solutions And Properties Of Nonlinear Partial Differential Equations Based On Symbolic Computation

The rapid development of nonlinear science has directly promoted the development of mathematical physics related fields,making it a maj or development in the 20th century after the quantum mechanics and relativity.Solitons are one of the first nonlinear phenomena that can be observed in nature and simulated in the laboratory.As an important branch of nonlinear science,soliton theory research not only opens up new research fields of mathematical physics,but also has important applications in many high-tech fields.The soliton theory has attracted more and more attention from researchers in nonlinear science,and has gradually become a research hotspot in mathematical physics.The main research content of this thesis is to study the soliton solutions of several nonlinear partial differential equations by using Hirota technology,B(?)cklund transformation and linear superposition principle.Based on these methods,we solves the soliton solutions of some nonlinear partial differential equations by using Maple's symbolic calculation and analyzes the correlation properties of soliton solutions.In this thesis,the bSK equation is taken as the research object,and its soliton solution,B(?)cklund transformation and interaction solution are studied by the Hirota bilinear method.Different from the existing research results,based on the bilinear form of Hirota,firstly,we use Maple's symbolic computing function to obtain one wave solution,two wave solution and three wave solution.In order to study the nature and interaction of solitons,We have chosen the appropriate parameters to make the solitons collide and separate from each other when drawing the motion process of multi-wave solutions with Matlab.Then we constructed the B(?)cklund transformation of the bSK equation based on the bilinear form of Hirota.Finally,eight types of lump-kink solutions are obtained by adding the exponential term to the lump solution.The lump-kink solution is a solution generated by the interaction of lump solitons and kink waves.In order to study the dynamics of the lump-kink solution more intuitively,we obtain the lump soliton from the interaction solution to study the motion of the lump soliton,and use Maple's powerful symbolic computing function to obtain the parametric equation of the lump soliton motion trajectory.We also confirmed that the lump term of the lump-kink solution is a class lump solution of the bSK equation.In-depth study of the lump term is beneficial to observe the collision process between the lump soliton and the kink wave.In addition,we construct the resonance multi-wave solutions of a(3+1)-dimensional generalized water wave equation,a new KP-like equation and a(3+1)-dimensional composite BKP equation based on their Hirota bilinear equation by using the linear superposition principle of exponential traveling wave solution.Resonance multi-wave solutions can help us study the resonance phenomena described by nonlinear partial differential equations.Finally,we find that the resonant multi-wave solutions of the three models have similar shapes by selecting the appropriate parameters and plotting the image of the resonant multi-wave solution.In this thesis,the soliton solutions of several kinds of nonlinear partial differential equations are constructed based on the Hirota bilinear technique,the B(?)cklund transformation and the linear superposition principle.In the process of research,some new forms of solutions were obtained.We have achieved certain research results by observing the interaction process of multiple solitons and the properties of soliton collisions.However,research also has problems such as the lack of application of new technologies and the complexity of the solution process.