Coherent structures for breather-soliton molecules and breather molecules of the modified KdV equation

The real modified Korteweg-de Vries equation governs the modulation of weakly nonlinear waves. We first review the multiple soliton solutions to the mKdV equation by means of the inverse scattering method in detail. It is found the soliton solutions are related to pure imaginary discrete eigenvalues, while the breathers are derived from complex eigenvalues. A novel expression for the mulitple soliton solution is presented which is used to construct the soliton and breather solutions. By introducing resonance condition for solitons and breathers, some resonant structures for breathers and solitons, or soliton bound states are first constructed for the real mKdV equation, such as breather molecules, breather-soliton molecules. Our work demonstrates the interactions among breather molecules and breather-soliton molecules are nonelastic by the meaning the breathers and solitons change their sizes.