Nonlinear Wave Solutions And Their Dynamics Of Fujimoto-Watanabe Series Equations With Quartic Nonlinearity

The study of nonlinear waves of differential equations and their dynamics has always been an important research fields in modern mathematics physics.In this thesis,we study the bifurcations and dynamics of traveling wave solutions to Fujimoto-Watanabe series equations with quartic nonlinearity from the perspective of dynamic systems.Because of their complex nonlinear structure(quartic nonlinearity),we exploit certain techniques,such as transformation,integration and multiplied by a factor,to transform them into planar dynamical systems,according to their specific structures.Then we obtain the bifurcation conditions and all possible bifurcations of phase portraits of the systems in different regions of the parametric space by using the qualitative theories and bifurcation methods,from which we show the sufficient conditions to guarantee the existence of traveling wave solutions including solitary wave solutions,periodic wave solutions,compactons and kink-like(antikink-like)wave solutions.We obtain their exact expressions by exploiting the first integral of planar system to integral on some special orbits,such as homoclinic orbits,heteroclinic orbits and periodic orbits,and we further investigate their dynamic properties.These results will help us understand the physical structure and propagation of the nonlinear wave.The main structure is as follows.The first part of the thesis mainly introduce the basic knowledge of plane dynamic system,the research methods and research background and existing results of Fujimoto-Watanabe equations with quartic nonlinearity.In the second part to the fifth part,we study the bifurcations and dynamics of traveling wave solutions to four Fujimoto-Watanabe equations with quartic nonlinearity respectively,and present the corresponding results and the proofs.The sixth part mainly summarizes the main research results of this thesis,and makes a prospect for the future research.