Research On Traveling Wave Solutions Of Two Classes Of Nonlinear Partial Differential Equations

Traveling wave solutions(TWS)play vital roles in the study of NPDEs.From the mathematical point of view,TWS can be well used to describe the long time behaviour of a nonlinear partial differential equation.From the physical point of view,TWS is helpful to understand the complicated nonlinear wave phenomenon and wave propagation.In this paper,we study the traveling wave solutions of two kinds of nonlinear partial differential equations.One class of equations is the(3+1)-dimensional Kadomtseve-Petviashvili-Boussinesq(KP-Boussinesq)equation,and the other class is the(2+1)-dimensional breaking soliton equation,they all have important applications in physics.The KP-Boussinesq equationcan was used to describe the growth of quasi-one dimensional shallow water waves when the impact of surface tension and viscosity are minimal and is widely applied in various physics fifields such as the internal and surface oceanic waves,ferromagnetics,nonlinear optics,Bose-Einstein condensation.The(2+1)-dimensional breaking soliton equation equation was used to describe the(2+1)-dimensional interaction of a Riemann wave propagating along the y-axis with a long wave along the x-axis.It is just the fact that makes the bifurcation theory of dynamical system become a powerful approach to investigate traveling waves of a PDE.Motivated by them,our strategy is to transform the traveling wave equation of(3+1)-dimensional KP-Boussinesq into a dynamical system in R~3.Fortunately,there exists a 2-dimensional invariant manifold which determines most of dynamical behaviours.Then,bifurcation analysis is applied to seek the parameter bifurcation sets which determine various qualitatively difffferent phase portraits.According to them,every orbit is identifified clearly and investigated in detailed including bounded and unbounded one.Finally,by calculating complicated elliptic integrals along these orbits,we obtain analytic expressions of all traveling wave solutions of the(3+1)-dimensional KP-Boussinesq equation without any loss.The obtained solutions well complement the types of traveling wave solutions of the(3+1)-dimensional KP-Boussinesq equation and are helpful to understand the complicated nonlinear wave phenomena and wave propagation,as well as help to construct more exact solutions of this equation including the multi-soliton solutions.The same research method was applied to(2+1)-dimensional breaking soliton equation,by using the relevant results of the KP-Boussinesq equation,we obtain the exact expressions of all its bounded traveling wave solutions and their numerical simulations.