Study On The Fornberg-Whitham Equation With Weak Dissipation And Dispersion Disturbance

The Fornberg-Whitham equation is a shallow water wave model,in such models,whether the equation has soliton solution and wave breaking phenomenon is of great significance.In this paper,we first study the local well-posedness and the blow up of the solutions for the Fornberg-Whitham equation with weak dissipation;Secondly,we study the soliton solution of the dispersion perturbed Fornberg-Whitham equation and the influence of the perturbed coefficient on the solution of the Fornberg-Whitham equation.In Chapter 3,the Cauchy problem of the weakly dissipative Fornberg-Whitham equation is studied.By using the prior estimation and energy inequality,the existence and uniqueness of the solution of the initial value problem are obtained according to the Banach fixed point theorem.In Chapter 4,we study the blow up phenomena of the solutions of the weakly dissipative Fornberg-Whitham equation.The blow up criteria for the solution of the weakly dissipative Fornberg-Whitham equation are given by using the method of inverse proof,and then the conditions for the initial value to be satisfied at the time of blow up are given.Finally,the blow up rate and the properties of blow up point set are discussed.The results show that the blow up rate of FW equation has nothing to do with the weak dissipation term,but the blow up condition of FW equation is affected.In Chapter 5,we study the traveling wave solutions of Fornberg-Whitham equation with perturbed dispersion?u_xxx.By using the bifurcation theory of dynamical system,the implicit smooth soliton solution,the explicit peakon and the periodic cusp wave solution are obtained,and it proves that the limit of the smooth soliton solution and the periodic cusp wave solution is the peakon.When the perturbed coefficient?tends to zero,the perturbed dispersion Fornberg-Whitham equation tends to the Fornberg-Whitham equation,the traveling wave solution obtained is consistent with the Fornberg-Whitham equation.