The Analysis Of The Solutions For A Class Of Shallow Water Wave Equations

In this paper, we study the stability of peakons for the Degasperis -Procesi equation with dispersion, and well-posedness of the Cauchy problem for the viscous weakly dispersive Degasperis-Procesi equation with initial data. The D-P (Degasperis-Procesi, i.e., DP equation) equation is deprived by Degasperis and Procesi. It has not only peaked but also shock peakons. They found that there are only three equations which satisfy the asymptotic integrability condition in this family: the Kdv equation, the Camassa-Holme equation and the Degasperis-Procesi equation. Thus they have similar properties. In the field of the fluid, when the shallow water wave which expressed by the Degasperis-Procesi equation occurs dispersion, the dispersion waves model can be described as Degasperis-Procesi equation with dispersion. It has smooth solitons, the peaked solitons and the period cuspons.In Chapter Three, we study the oribal stability of peakons for the Degasperis- Procesi equation with dispersion by variational approach that constructs the solitary waves as energy minimizers under appropriate constraints. Then using of the stability of the conclusion, through appropriate transformation, we can convert the problems of DP equation for the generalized nonvanishing boundary peaked solutions' stability to stability of peaked solution for Degasperis-Procesi equation with dispersion. In Chapter Four, since we are interested in the effect of the weakly dispersive term to the viscous Degasperis-Procesi equation, using the Kato's theory, we obtain the local well-posedness of the Cauchy problem for the viscous weakly dispersive Degasperis-Procesi equation.