Multiple-pole solutions and degeneration of breather solutions to the focusing nonlinear Schrödinger equation

Abstract

Based on Hirota’s method, the multiple-pole solutions of the focusing Schrödinger equation are derived directly by introducing some new ingenious limit methods. We have carefully investigated these multi- pole solutions from three perspectives: rigorous mathematical expressions, vivid images, and asymp- totic behavior. Moreover, there are two kinds of interactions between multi-pole solutions: when two multiple-pole solutions have different velocities, they will collide for a short time; when two multiple- pole solutions have very close velocities, a long time coupling will occur. The last important point is that this method of obtaining multiple-pole solutions can also be used to derive degeneration of N-breather solutions. The method mentioned in this paper can be extended to derivative Schr¨odinger equation, Sine-Gorden equation, mKdV equation and so on.