Study On Singular Travelling Wave Solutions Of The Nonlinear Evolution Equations

The mathematical modeling of enormously important phenomena arising in physics, mechanics, biology and other research fields often leads to nonlinear equations which are usually named nonlinear evolution equations. It is remarkable that many of these nonlinear equations possess a regular behavior, typical of integrable partial differential equations, that is there possess Hamilton structures.In 1993, Camassa and Holm derived a completely integrable dispersive shallow water equation by using Hamiltonian methods and showed a soliton solution for this equation has a limiting form that has a discontinuity in the first derivative at its peak. In the same year, to understand the role of nonlinear dispersion in pattern formation, Rosenau and Hyman introuduce and study Korteweg-de Vries-like equation K(m,n) with nonlinear dispersion and presented their solitary wave solutions has compact support (which was named compacton), that is they vanish identically outside a finite core region. The appearences of these non-smooth travelling waves by propagation of the smooth initial conditions attracted many research attentions. Thse topics have seen significant advances and research is also very active.We consider the travelling wave solution of these partial differential equations which are reduced into ordernary differential equations with travelling wave speed c as a parameter and a PDE with Hamiltonian structures into a integrable ODE which can be studied by using of the bifurcation theores of the dynamical system and thus understand their dynamics for some classes of the classical nonlinear evolution equations.The aim of this dissertation is to study and understand the dynamics.of the traveling wave soltions of nonlinear wave equations, such as smooth periodic wave solutions, solitarys, kink waves and other non-smooth wave solutions such as compacton, cuspon and periodic cusp waves, et al and with a special emphasis on singular wave.In this dissertation, we emplied the bifurcation theory of the dynamical systems and the transformation techniques to study the traveling wave soltions of several classes of extensively used classical nonlinear wave equations including the generalized Camassa-Holm equation, Degasperis-Procesi equation, generalized B(m,n) equation with nonlinear dispersion, generalized Schrodinger eqution, generalized K(m,n,k) equation and generalized Klein -Gordon model equation and their bifurcations with an emphasis on singular waves. It is worthy to point out that we present two types of singular waves which are different to the common known ones. One will see from this dissertation that the bifurcation theory of the dynamical systems and the transformation techniques can be used not only to clarify the reason of the existences of these common known singular waves but also to find some other typies ones.