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This article concerns an approximate solution for the Benjamin-Bona-Mahony (BBM) equation with a bilinear control in slowly varying medium. By a sharp estimation of the error term, a suitable approximate solution for this equation is established.

We study the inelastic collision of two solitary waves of different velocities for the generalized Benjamin-Bona-Mahony (BBM) equation with cubic nonlinearity. It shows that one solitary wave is smaller than the other one in the H^1(R) energy space. We explore the sharp estimates of the nonzero residue due to the collision, and prove the inelastic collision of two solitary waves and nonexistence of a pure 2-soliton solution.

In this article, we study a generalized weakly dissipative periodic two-component Camassa-Holm system. We show that this system can exhibit the wave-breaking phenomenon and determine the exact blow-up rate of strong solution to the system. In addition, we establish a sufficient condition for having a global solution.