A Vector General Nonlinear Schrödinger Equation with (m+n) Components

A vector general nonlinear Schrodinger equation with $$(m+n)$$ components is proposed, which is a new integrable generalization of the vector nonlinear Schrodinger equation and the vector derivative nonlinear Schrodinger equation. Resorting to the Riccati equations associated with the Lax pair and the gauge transformations between the Lax pairs, a general N-fold Darboux transformation of the vector general nonlinear Schrodinger equation with $$(m+n)$$ components is constructed, which can be reduced directly to the classical N-fold Darboux transformation and the generalized Darboux transformation without taking limits. As an illustrative example, some exact solutions of the two-component general nonlinear Schrodinger equation are obtained by using the general Darboux transformation, including a first-order rogue-wave solution, a fourth-order rogue-wave solution, a breather solution, a breather–rogue-wave interaction, two solitons and the fission of a breather into two solitons. It is a very interesting phenomenon that, for all $$M>0$$, there exists a rogue-wave solution for the two-component general nonlinear Schrodinger equation such that the amplitude of the rogue wave is M times higher than its background wave.

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