The Study On Inelastic Collision Of Two Solitary Waves For The Generalized BBM Equation

It is known that many nonlinear models can be used to describe the physical,chemical and biological phenomena. The last few decades have seen an enormous growth in the applicability of nonlinear models and in the development of related nonlinear concepts. This has been driven by modern computer power as well as by the discovery of new mathematical techniques, which include three contrast themes: the first one is the study of chaos, the second one is the study of fractals, and the last one is the study of solitons. However, not all systems arising from physical phenomena are integrable, for example, generalized Benjamin-Bona-Mahony equation (simply for gBBM). Applications of nonlinear models range from atmospheric science to condensed matter physics and to biology, from the smallest scales of theoretical particle physics up to the largest scales of cosmic structure. Problems related to dynamical behavior of solitons have fascinated applied mathematicians, physicists and engineers for a long time.In this paper, employing the method of approximate solution, we mainly study the inelastic collision of two solitary waves for gBBM equation under two special cases.One is the study on inelastic collision of two solitary waves of nearly equal speed for the gBBM equation. The other one is the study on inelastic collision of two solitary waves of different speed for the gBBM equation. The main arguments for these two cases are the construction of approximate solutions in the collision regime. We now sketch the main steps of the proofs for these two cases.Case one:First, we construct an approximate solution to the problem in the collision regime and point out the algebra structure of the approximate solution. After the approximate solution is constructed, we introduce a delicate decomposition of the solution. Finally,by controlling the error term in the approximate solution, symmetric arguments and proof by contradiction, we prove the inelastic character of the collision of two solitary waves.Case two:First, we construct an approximate solution to the problem in the collision regime and show the delicate algebra of the approximate solution which is different from the case one. Second, using asymptotic arguments, we justify that the solution of the gBBM equation is close to the approximate solution and we control the solution in large time.Finally,we prove the inelastic character of the collision of two solitary waves with a further analysis of the approximate solution.By studying on the inelastic collision of the two solitary waves for gBBM equation,we prove that the two-soliton structure is stable globally and nonexistence of a pure 2-soliton solution after the collision. It is found that the defect is due to a non-zero extra term in the approximate solution after recomposition of the series.