Explicit solutions for a semidiscrete Boussinesq system

A semidiscrete Boussinesq system has been introduced by the means of recombination flows from a 4×4 spectral problem. We construct the Darboux transformation and explicit solutions for the semidiscrete Boussinesq system. We can find that the 1-fold Darboux-transformation solution of the system possesses more than one traveling waves. Especially, one of solutions comes into being two solitary waves after a period of time, which means that properties of solutions to semidiscrete integrable systems are richer than the ones in continuous integrable systems.