A new vector long wave‐short wave‐type model is proposed by resorting to the zero‐curvature equation. Based on the resulting Riccati equations related to the Lax pair and the gauge transformations between the Lax pairs, multifold Darboux transformations are constructed for the vector long wave‐short wave‐type model. This method is general and is suitable for constructing the Darboux transformations of other soliton equations, especially in the absence of symmetric conditions for Lax pairs. As an illustrative example of the application of the Darboux transformations, exact solutions of the two‐component long wave‐short wave‐type model are obtained, including solitons, breathers, and rogue waves of the first, second, third, and fourth orders. All the solutions derived by the Darboux transformations involve square roots of functions, which is not observed in the investigation of other nonlinear integrable equations. This model describes new nonlinear phenomena.

中文翻译：

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向量长波短波模型

借助零曲率方程，提出了一种新的矢量长波短波类型模型。基于由此产生的与Lax对有关的Riccati方程以及Lax对之间的规范变换，为矢量长波-短波类型模型构造了多个Darboux变换。此方法是通用的，适用于构造其他孤子方程式的Darboux变换，尤其是在没有Lax对对称条件的情况下。作为应用Darboux变换的一个说明性示例，获得了两分量长波短波类型模型的精确解，包括一阶，二阶，三阶和四阶的孤子，通气和无赖波。由Darboux变换得出的所有解都包含函数的平方根，在研究其他非线性可积方程时没有观察到这一点。该模型描述了新的非线性现象。