Symbolic Computation On Integrability And Soliton Solutions For The (2+1)-dimensional Zakharov-Kuznetsov Equations And Normalized Linearly Coupled Nonlinear Wave Equations

Nonlinear science is a symbol of modern science. As a branch of the nonlinear science, soliton has been applied in mathematics, hydrodynamics, plasma, nonlinear optics and other fields. Soliton solutions are a special kind of solutions for nonlinear partial differential equations (NPDEs) and possess cenain physical significance when the equations represent some physical processes. Meanwhile, with the development of the soliton theory, some analytical methods have been presented to investigate the analytic solutions and various properties of the NPDEs such as the Backlund transformation, nonlinear superposition and Lax pair. Those methods contain the inverse scattering method, Backlund transformation, Darboux transformation, Hirota direct method, Wronskian technique and Painleve test. accurately. The mathematic calculational and graphical functions of symbolic computation software can help us to perform the analytical and observable study on the analytical solutions and properties of the NPDEs, so that it provides an assistant tool for investigating the NPDEs.Using Hirota direct method, Wronskian technique and Painleve test, we study soliton solutions, Backlund transformation, Painleve property and Lax pair. The structure of the present dissertation is organized as follows:In Chapter 1, we first introduce the history and development of the soliton. Then we explain three methods—Painleve analysis, Hirota direct method and Wronskian technique.Chapter 2 is aim to (2+1)-dimensional Zakharov-Kuznetsov (ZK) equation in the electron-positron-ion plasmas. Firstly, we can construct the N-soliton solution in the form of exponentials by introducing the proper transformation. The dynamic properties of soliton collisions will be illustrated graphically in detail. Meanwhile, parametric analysis is carried out in order to illustrate that the soliton amplitude and width are affected by the phase velocity, ion-to-electron density ratio, rotation frequency and cyclotron frequency.Chapter 3 is devoted to the N-soliton solution in Wronskian of (2+1)-dimensional ZK equation in the electron-positron-ion plasmas. In contrast to the N-soliton solutions derived from the bilinear method and the inverse scattering method, the N-soliton solution in Wronskian has better differential properties so that it can be verified by direct substitution of the solution into the bilinear equations using the symbolic computation. In this chapter, we construct the N-soliton solution in Wronskian for (2+1)-dimensional ZK equation. And verify the validity of the solution by virtue of properties of the Wronskian and Laplace theorem.Chapter 4 is aim to (2+1)-dimensional variable-coefficient ZK equation in plasmas. Firstly, via Painleve analysis, we get that the equation does not possess Painleve integrability. After that, we can construct the N-soliton solution in the form of exponentials by introducing the proper transformation. The dynamic properties of soliton collisions will be illustrated graphically in detail.In Chapter 5, analytically investigated are the normalized linearly coupled nonlinear wave equations in the two-core optical fiber and Caudrey-Dodd-Gibbon equation. The one-and two-soliton solutions are obtained via the Hirota's method and symbolic computation. Backlund transformation is given, and Lax pair is derived.