Research On Analytical Solutions And Dynamic Behavior Of Several Nonlinear Wave Equations

In this work,we mainly investigate several types of nonlinear wave equations by Hirota bilinear method and Darboux transformation.Two types of new nonlinear evolution equations are derived from hydrodynamics model via the standard reductive perturbation theory.At the same time,the dynamic behavior of the obtained solutions are analyzed through image simulation.The main contents of this paper are as follows:In chapter 1,the research background and significance of soliton theory and related methods are briefly introduced,and the main contents of this paper are given.In chapter 2,the analytical solutions of two types of nonlinear wave equations are mainly investigated.First,the lump solutions and lumpoff solutions of(2+1)-dimensional Hirota-Satsuma-Ito(HSI)equation are successfully constructed via the Hirota bilinear method.When the lump soliton interacts with an exponentially twin plane soliton,the rogue waves are obtained.The relevant appearance time and place are predicted accurately.Then,based on the bilinear expression of the Zakharov-Kuznetsov(ZK)equation,soliton solutions are obtained explicitly.By adjusting the spectrum parameters,we can obtain the breather wave solutions and general hybrid wave solutions.In addition,these significant characteristics of nonlinear waves are illustrated to better understand their dynamical behavior.In chapter 3,we mainly investigate two types of plasma models.In the dusty plasma model and quantum hydrodynamic(QHD)model,a(2+1)-dimensional and a(3+1)-dimensional nonlinear wave solutions are derived by the reduced perturbation method.Based on the bilinear formalism of these two equations,abundant analytical solutions are constructed,such as Grammain solutions,soliton solutions,breather wave solutions,lump solutions,resonance solutions and semi-rational solutions.It is interesting that the propagation characteristics of these waves are depended on the value of the plasma parameters.In chapter 4,based on the classic Darboux transformation,we construct the Darboux-dressing transformation of coupled Gerdjikov-Ivanov(cGI)equation by the perturbation expansion method.That is,we iterate with the same spectral parameters to find exact solutions which reduce the computational complexity.By means of separating variables,the vector breather wave solutions are derived.Moreover,higher-order rouge wave solutions are presented via Taylor multi-series expansion method.In chapter 5,brief summaries of whole paper and related prospects for future research work have been shown.