Existence And Explicit Expression Of Bounded Traveling Wave Solutions For Coupled KdV Type Equations

Research on nonlinear evolution equations bounded traveling wave solutions cannot only help understand the nature of the properties of soliton theory, but also play animportant role to offer a reasonable explanation of natural phenomena. Therefore, thenonlinear evolution equations bounded traveling wave solutions has become a majorresearch topic in different branches of the mathematical and physical sciences, such asphysics, biology, chemistry, optical communications.This paper mainly use theory and methods of planar dynamical systems,undetermined coefficient method, function expansion method, as well as ?GGExpands method to research bounded traveling wave solutions of nonlinear evolutionequations, for example:For the equation(I), firstly make line of the traveling wave transform, turn it intoits equivalent planar dynamical systems, analyze limited singular point by the planardynamical systems theory and method, draw the phase diagram and the rail linedistribution of equivalent planar dynamical systems with the Maple mathematicalsoftware. According to the equivalent planar dynamical system with equation(I),Using function expand method and ?GG Expands method, gain the conclusions thatequation(I) has a bell-shaped solitary wave solutions, a periodic solutions and fourbounded traveling wave solutions. Four bounded traveling wave solutions are moregeneral, solutions derived from previous literature can be a corollary of this article.The coupled equation(II) with nonlinear cubic item, firstly make line of thetraveling wave transform, the equation into equivalent planar dynamical system, byusing the theory of planar dynamical systems and the method to be limited in thedistance the singularity analysis, which the coupled nonlinear equations(II) of theplanar dynamical system under the different parameters of phase diagram are gaven,according to the phase diagram and the trajectory distribution, that the existence ofbounded traveling wave solutions of the conditions of equations(II).And we used themethod of undetermined coefficients, equation(II) the three bell-shaped solitarysolutions and one kink solitary solitary wave solutions of an explicit expression. Thesolutions cannot be deduced from the former references.