New types of multisoliton solutions of some integrable equations via direct methods

Exact explicit solutions, which describe new multisoliton dynamics, have been identified for some KdV type equations using direct methods devised for this purpose. It is found that the equations, having multi-soliton solutions in terms of the KdV-type solitons, possess also an alternative set of multi-soliton solutions which include localized static structures that behave like (static) solitons when they collide with moving solitons. The alternative sets of solutions include the steady-state solution describing the static soliton itself and unsteady solutions describing mutual interactions in a system consisting of a static soliton and several moving solitons. As distinct from common multisoliton solutions those solutions represent combinations of algebraic and hyperbolic functions and cannot be obtained using the traditional methods of soliton theory.