Long-Range Interaction between Two Periodic Solitons through Growing-and-Decaying Mode in the Davey–Stewartson I Equation

where r is constant. One important feature of the solutions to the DS I equation is the reversion of the unstable wave train to its initial state. The analytical solution for the nonlinear evolution of a modulational unstable mode is given by the growing-and-decaying (GD) mode solution. This shows that an unstable mode grows exponentially in its early stage. After reaching a maximummodulation, the unstable mode vanishes with time to reproduce the initial unmodulated state. The DS equation has a periodic soliton solution, which represents the spatial structure of the inclined sequence of algebraic solitons in addition to the dark line soliton and algebraic soliton solutions. In previous studies, we investigated the interactions between two periodic solitons, and between the periodic soliton and other types of solitons. Note that there are two types of singular interaction: the resonant interaction, in which two solitons interact so as to generate a single soliton, and the extremely long-range interaction, in which two solitons interchange each other infinitely apart through the messenger soliton, which is similar to a periodic soliton. Recently, the interaction between a soliton and a GD mode has been investigated, and the existence of resonant interactions has been demonstrated. We reported that the existence of the periodic soliton changed the evolution of the modulational instability markedly, as if the soliton dominated the evolution of instability. These results suggest the possible existence of an interaction between two solitons through the GD mode. The purpose of the present note is to show that a long-range interaction exists between two parallel periodic solitons through the GD mode. Using the N-soliton solution of Satsuma and Ablowitz, the solution that describes the interaction between two periodic solitons, which have the same real part ð ; Þ and different imaginary parts ð 1; 1Þ and ð 2; 2Þ of complex wave numbers, is written as