Study On Properties Of Solutions For A Class Of Nonlinear Wave Equations

In this paper,we consider properties of solutions for Kuramoto-Sivashinsky equation,damped wave equation and coupled-mode equations.These equations are widely used in flame waves,thin film vibration,the optical fiber communication and other problems.By using Green function method and priori estimates,we obtain the existence and uniqueness of global classical solutions and its decay estimate of the first two equations;By using hyperbolic function expansion method,some traveling wave solutions of the third equation are obtained and its physical significance is discussed.This paper is organized in five chapters.In the first chapter,we give the physical background for the nonlinear evolution equations,such as Kuramoto-Sivashinsky equation,damped wave equation and coupledmode equations.We look back to some significant results,then give our results.In the second chapter,we consider the Cauchy problem of the generalized KuramotoSivashinsky equation in multi-dimensions.The difficulty is that KS equation consists of higher order derivatives.To overcome this difficulty,we give the Green function and the energy estimates in section 2.2.In section 2.3,we got the global classical solutions to the equations by means of constructional iterative equation.Then we obtain the pointwise estimate of solution by some priori estimates.In the third chapter,we consider the Cauchy problem of the damped wave equation.We first divide the solution into two parts by long wave-short wave decomposition.Then we get the estimate of long wave part by Green function method and short wave part by energy method.Similarly,we get the global classical solutions to the equations by means of constructional iterative equation.At last,L~P(2 ?p??)estimate is obtained by priori estimates.In the fourth chapter,we consider the traveling wave solutions of coupled-mode equations.In section 4.2,when?=0,we utilize the hyperbolic function expansion method and homogeneous balance method to obtain some special traveling wave solutions,including kink soliton,anti-kink soliton and dark soliton.In section 4.3,when??0,we construct an auxiliary function to get other traveling wave solutions.In the fifth chapter,we summarize our works and point out the further research directions.