Long-range interaction between two solitons through modulational instability: Solution of the Davey-Stewartson equation

Abstract:
A weak nonlinear uniform water wave train is unstable to modulational perturbations of long wave length. The long time evolution of a two-dimensional wave-packet is described by the Davey-Stewartson (DS) equation.   One of important features of the solutions to the DS equation is the reverting of the unstable wave train to its initial state. A growing-and-decaying mode solution to the DS equation is one of the recurrent solutions. The wave grows exponentially according to the linear instability at the initial stage, reaches a state of maximum modulation after some finite time and returns to the unmodulate initial state.

We show the existence of lone-range interaction between two periodic solitons through the growing-and-decaying mode. When two periodic solitons approach each other some distance, the growing-and-decaying mode starts to grow in the space between two solitons and then solitons exchange the transverse wave numbers.

We will also discuss the interaction between two line solitons through the grwoing-and-decaying mode.