A vector general nonlinear Schrödinger equation with \((m+n)\) components is proposed, which is a new integrable generalization of the vector nonlinear Schrödinger equation and the vector derivative nonlinear Schrödinger 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 Schrödinger 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 Schrödinger 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 Schrödinger equation such that the amplitude of the rogue wave is M times higher than its background wave.

中文翻译：

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具有$$（m + n）$$（m + n）分量的向量一般非线性Schrödinger方程

提出了具有（（m + n）\）分量的矢量广义非线性Schrödinger方程，它是矢量非线性Schrödinger方程和矢量导数非线性Schrödinger方程的一种新的可积分推广。借助于与Lax对关联的Riccati方程和Lax对之间的规范变换，构造了具有\（（m + n）\）分量的向量一般非线性Schrödinger方程的一般N倍Darboux变换，可以直接简化为经典N倍数Darboux变换和广义Darboux变换而不受限制。作为说明性示例，通过使用一般的Darboux变换获得了两成分的一般非线性Schrödinger方程的一些精确解，包括一阶流浪解，四阶流浪解，通气解，通气-流浪相互作用，两个孤子和通气分裂为两个孤子。一个非常有趣的现象是，对于所有\（M> 0 \），对于二分量一般非线性Schrödinger方程都存在一个流浪解，使得流浪的振幅比其背景波高M倍。 。