Darboux transformation and classification of solution for mixed coupled nonlinear Schrödinger equations

• The compact expressions of solutions are obtained by Darboux transformation.
• The conditions of classification of solution are given by the matrix analysis method.
• The rogue wave, breather and dark soliton solutions are obtained.
• The multi-localized wave solutions are given in detail.

We derive generalized localized wave solution formula for mixed coupled nonlinear Schödinger equations (mCNLSE) by performing the unified Darboux transformation. Based on the dynamical behavior of solution, the classification of the localized wave solutions on the nonzero background is given explicitly. Especially, the parameter conditions for breather, dark soliton and rogue wave solution of mCNLSE are given in detail. Moreover, we analyze the interaction between dark soliton solution and breather solution. These results would be helpful for nonlinear localized wave excitations and applications in vector nonlinear systems.