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Bounded states for breathers–soliton and breathers of sine–Gordon equation

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Abstract

The Wronskian solutions to the sine–Gordon (sG) equation that can provide interaction of different kinds of solutions are revisited. And a novel expression of N-soliton solution with a nonzero background to the sG equation is presented. The kinks (solitons), the breathers and the interactions among solitons and breathers are also derived directly from the novel expression. 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 breather–soliton molecules (BSMs) and the bounded state for breather molecules (BMs). An approximately bounded state for kinks (solitons) is given for the wavenumber being nearly the same. In addition, it is demonstrated that the interactions among the BSMs, BMs, solitons and breathers may be inelastic by the particular meaning the sizes of the BSMs and BMs change.

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Acknowledgements

The author is grateful to Prof. S. Y. Lou for helpful and enlightening discussion. The author also acknowledges the support of NNSFC (No. 11675084) and K. C. Wong Magna Fund in Ningbo University.

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  1. School of Physical Science and Technology, Ningbo University, Ningbo, 315211, People’s Republic of China

    Man Jia

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Jia, M. Bounded states for breathers–soliton and breathers of sine–Gordon equation. Nonlinear Dyn 105, 3503–3513 (2021). https://doi.org/10.1007/s11071-021-06799-0

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