Bounded States for Breathers–Soliton and Breathers of Sine–Gordon Equation

Abstract The Wronskians solutions to the sine-Gordon (sG) equation that can provide interaction of different kinds of solutions are revisited. A novel expression N-soliton solution with a nonzero background is presented which is used to construct the soliton and breather solutions. Due to the existences of abundant structures of the solitons and breathers, it is possible to search for the coherent structures, or bounded states of solitons and breathers. By introducing the velocity resonant conditions, the sG equation is proved to possess the bounded state for breathers-soliton, or breather-soliton molecules (BSMs) and the bounded state for breathers, or breather molecules (BMs) by different parameters selections. In addition, an approximately bounded state for solitons is demonstrated. The interesting thing is the interactions among the BSMs, BMs and solitons, breathers may be nonelastic by the particular meaning the sizes of the BSMs and BMs change.