Asymptotic Properties Of Solutions For Nonlinear Telegraph Equations And Exact Travelling Wave Solutions For Generalized KdV Equations

The objectives of this work are twofold. Firstly, we apply asymptotic the-ory and perturbation methods to develop the well-posedness of solutions for twokinds of nonlinear perturbation telegraph equations. Secondly, we aim to acquirethe travelling wave solutions for several types of generalized KdV equations byusing the mathematical technique based on the reduction of order for solvingdi?erential equations.Chapter 2 deals with the well-posedness of solutions for the initial valueproblem of semilinear telegraph equations utt ? uxx + u +εh(u,ut,ux,ε) = 0 inan infinite region {(x,t)|x∈R,0 < t < L/ |ε|}. The asymptotic validity offormal approximations for the solutions in the region is established. An appli-cation for the asymptotic theory obtained is given at the end of this chapter.In chapter 3, we study the global existence of solutions for initial value prob-lems of semilinear telegraph equations utt ?uxx +2aut +2bux +cu =εf(u,ε) inone space dimension. By making use of the Banach's fixed point theorem, thewell-posedness of solutions is established in the classical sense of C2.In chapter 4, using a mathematical technique based on the reduction oforder for solving di?erential equations, we study two types of potential mKdVequations in (1+1) dimensions and a potential mKdV equation in (2+1) di-mensions, which are ut + a(ux)2 + buxxx = 0, ut +αupux +βuxxx = 0 and (ut + a1(ux)2 + b1uxxx)x + kuyy = 0. The analytical expressions of the travellingwave solutions for the three equations are derived. For the (1+1) dimensionalmKdV equation with positive or negative exponents, it is shown that the expo-nent of the wave function u together with the ratio between the wave speed andthe variant coe?cient of the highest di?erential term in the equation determinesthe physical structures of the solutions.In chapter 5, we acquire the exact travelling wave solutions for general-ized two-dimensional KdV-type equation ut + aux + b(un)x + b(un)y + (un)xxx +(un)yyy = 0, where n > 0 or n < 0. The compactons, solitons, solitary patternsand periodic solutions for the equation are obtained.