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

Based on the Hirota's method, the multiple-pole solutions of the focusing Schrodinger 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 asymptotic behavior. Moreover, there are two kinds of interactions between multiple-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 the degeneration of N-breather solutions. The method mentioned in this paper can be extended to the derivative Schrodinger equation, Sine-Gorden equation, mKdV equation and so on.