Constructing new periodic exact solutions of evolution equations.

Abstract

For the nonlinear Schrödinger equation, the Korteweg-de Vries equation, and the modified Korteweg-de Vries equation, periodic exact solutions are constructed from their stationary periodic solutions, by means of the Bäcklund transformation. These periodic solutions were not written down explicitly before to our knowledge. Their asymptotic behavior when t-->-infinity is different from that when t-->infinity. Near t=0, the spatial-temporal pattern can change abruptly, and rational solitons can appear randomly in space and time. They correspond to new types of "homoclinic orbits" due to different asymptotic behaviors in time.