Multi-soliton solutions and long-time asymptotic behavior of the modified Korteweg–de Vries equations

In this paper, two modified KdV (MKdV) equations, one is real and the other is complex, are investigated. By applying the Hirota bilinear operator theory and computer algebra, the corresponding bilinear forms of these two MKdV equations are successfully derived, and then their multi-soliton solutions are obtained and expressed in explicit forms. In order to study the dynamic properties of the obtained multi-soliton solutions, we further analyze their long-time asymptotic behaviors. It follows that the collisions between two solitons are elastic which are graphically shown.