Research On Several Problems Of Nonlinear Wave Equations

Nonlinear wave equations are important mathematical models for describing naturalphenomena and are one of the forefront topics in the study of mathematical physics,especially in the study of soliton theory. In this doctorial dissertation, we study sev-eral problems of nonlinear wave equations by using monotone dynamical systems theory,geometric singular perturbation theory, conservation laws and Sobolev spaces. Theseproblems include existence of travelling wavefronts, orbital stability of peakons and non-uniform continuity. The main work of this dissertation are as follows.In Chapter 2, we study the existence of travelling wavefronts for the KdV-Burgersequation and KBK equation. Motivated by the analogue between travelling wavefrontsand heteroclinic orbits of the corresponding ordinary di?erential equations, we firstlyobtain a su?cient condition for the existence of the KdV-Burgers equation by usingmonotone dynamical systems theory. It is worthwhile to note that the conditions underwhich some exact travelling wavefronts have been found are special cases of ours. Sec-ondly, using the geometric singular perturbation theory, we prove the existence of theKBK equation for su?ciently small dissipation. With the help of Matlab, the result ofnumerical investigation also establishes our analysis.In Chapter 3, we study the stability of peakons for the DP equation with strongdispersion. Since the conservation laws of the DP equation with strong dispersion aresame as the DP equation, using the method for studying the stability of peakons for theDP equation, we prove the stability of peakons for the DP equation with strong dispersion.It is worth noticing that not only our approach can be used for the Camassa-Holmequation with strong dispersion to gain more stability information, but also our resultcan be directly applied to study the stability of peakons with nonvanishing boundary forthe DP equation with a dispersive term.In Chapter 4, we show that the solution map of the periodic DP equation is notuniformly continuous in Sobolev spaces Hs(T) for s > 3/2. This extends previous resultfor s?2 to the whole range of s for which the local well-posedness is known. The proofis based on the method of approximate solutions and well-posedness estimates for theactual solutions.In Chapter 5, we show that the solution map of the periodic modified Camassa-Holmequation is not uniformly continuous in Sobolev spaces Hs(T) for s > 7/2. The proofis based on the method of approximate solutions and well-posedness estimates for theactual solutions.