Description Of The Nonexistence Of Pure Multi-solitons For The BBM Equation

In this paper, we consider the existence of pure multi-solitons for the Benjamin-Bona-Mahony(BBM) equation. To prove the result, we use the method of Martel and Merle, by estimating the tail of the solution for large t and large x, we obtain a contradiction. Indeed, we prove the nonexistence of pure multi-solitons for the BBM equation. Moreover, we consider the decay estimates for ingoing multi-soliton and construct an approximate multi-soliton solution for the BBM equation.There are three sections in this paper:In the first section, wo prove there is nonexistence of pure multi-solitons for the BBM equation. we consider the collision of two solitons. The method we used here is different from other papers. We consider the speed of solitons and use the conservation laws to control the speeds at the time that tends to infinity for an ingoing2-soliton. By reduction, we obtain the contradiction, thus, we prove there is nonexistence of pure N solitons.The second section is devoted to the decay estimates for ingoing multi-soliton.we introduce the famous Lyapunov function and obtain some monotonicity results.On this basis, we combine the properties of some function convolution and the expression of BBM equation to study the decay of solitons. Finally, we prove the solution of the BBM equation satisfies the property of exponential decay.In the third section, we construct an approximate multi-soliton solution for the BBM equation. We do this work mainly by three steps: first, we construct an approximate two soliton solution for the BBM equation. Second, we construct an approximate multi-soliton solution by the same method. We reserve the remainder term of the equation and make correction equations continuously to offset the major term of the equation, so we can get an approximate multi-soliton solution for the BBM equation. Finally, we estimate the remainder term of the equation and consider the property of the approximate multi-soliton solution.