The Study On The Approximate Controllability And Stability Of The Soliton Of GBBM Equation In Slowly Varying Medium

In this paper,we focus on the controllability and stability of the soliton of gBBM equation with internal bilinear control in slowly varying medium.1?_x²?_tu_x?_x²uumu²?f?t,x?,?,01?,wheref?t,x?n?t,x?u?t,x?.We adopt the method developed by Mu?oz when studying the approximate controllability and stability of the soliton of gKdV equation.We construct an approximate solution near the solitary wave of the equation firstly,and estimate the error term between the exact solution and approximate solution,it is proved that the error term can be controlled toO_H1?R?(²³e^?t?)by the Taylor formula,integral estimation,binomial theorem and truncated function in large time.By using the Lyapunov function,we obtain the stability result of the soliton of gBBM equation in slowly varying medium.In the first section,the background and significance,the research status and the content of this paper are introduced.In the second section,the relevant preliminary knowledge is given.In the third section,we construct an approximate solution near the solitary wave of gBBM equation in slowly varying medium,and estimate the error term between the exact solution and approximate solution,it is proved that the error term is controllable in large time.In the fourth section,the stability result of the equation by using the Lyapunov function is obtained,and two main theorems of the paper are proved.In the fifth section,we make the summary and outlook.