The Bifurcations Of Nonlinear Waves For D(m,n)system

In this dissertation, the nonlinear dispersion Drinfel’d-Sokolov system(simply calledD(m, n) system) are studied by the qualitative theory of di?erential equations and thebifurcation method of dynamical systems. Not only the bifurcation phases portraits ofthis system are given, but also various new traveling wave solutions of this system areobtained. Those solutions including solitary wave solutions, periodic wave solutions, kinkwave solutions, cusp wave solutions, singular wave solutions and trivial solutions. What’smore, some interesting bifurcations phenomena of D(m, n) system are found. At the sametime, the concrete variation of the traveling waves for those bifurcations phenomena areshown in the ?gures. The concrete research work of this dissertation as follows:In Chapter 1, ?rstly, we give a simple summary of the nonlinear wave equations andthe solving methods. Secondly, we introduce D(m, n) system and its results which weregiven by predecessors. Finally, we present the bifurcation method of dynamical systemsand the major works of this dissertation.In Chapter 2, we study the bifurcations of solitary waves and kink waves for D(1, 2)system. Firstly, by using the bifurcation method, we gain the bifurcation phases portrait-s, periodic wave solutions, solitary wave solutions and blow-up wave solutions of D(1, 2)system. Secondly, under the change of some parameters, we ?nd some bifurcation phe-nomena of D(1, 2) system as follows:(i) the fractional solitary waves can be bifurcatedfrom the trigonometric periodic waves and the elliptic periodic waves;(ii) The kink wavescan be bifurcated from the solitary waves and the singular waves. In the end, the concretevariation of those traveling waves are shown in the ?gure and the example con?rmed bysoftware Mathematica is given.In Chapter 3, we research the bifurcations of compact soliton and solitary cusp waves.Firstly, we give the bifurcation phases portraits, compact soliton, solitary wave solutions,solitary cusp wave solutions and singular cusp wave solutions of D(1, 2) system. Secondly,we show the bifurcation phenomena including the compact soliton can be bifurcated fromthe solitary waves, and the solitary cusp waves can be bifurcated from the solitary wavesand the singular cusp waves.In Chapter 4, we research D(2, 2) system. Firstly, we give the bifurcation phasesportraits and some exact traveling wave solutions, that is, periodic wave solutions, singu-lar periodic wave solutions and solitary wave solutions. Secondly, we prove the solitarywaves can be bifurcated from the singular periodic waves and the smooth periodic waves.Chapter 5 is the summary and prospect of this dissertation.