Quasi-solitons of the two-mode Korteweg-de Vries equation

The two-mode Korteweg-de Vries equation (TMKdV) was proposed to describe the propagation of nonlinear waves of two different wave modes simultaneously. However, the existence of multi-soliton solutions is still unknown. In this letter we present two Hamiltonians, the conservation laws, and a Miura-like transformation of the equation. We show that the TMKdV equation has quasi-soliton behaviour in which waves moving in the same direction pass through each other almost without change of their wave forms except for phase shifts.