A generalized [Formula: see text] matrix spectral problem is investigated to generate two five-component nonlinear integrable systems, which involve an arbitrary smooth function. These systems are proven integrable in the sense of Lax pair. As the reduction cases, a four-component reaction diffusion equation and a four-component modified Korteweg-de Vries (mKdV) equation are solved by Darboux transformation approach.

中文翻译：

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两个综合五分量非线性可积系统：Darboux 变换和应用

研究了一个广义的[公式：见正文]矩阵谱问题，以生成两个包含任意平滑函数的五分量非线性可积系统。这些系统在 Lax pair 的意义上被证明是可集成的。作为还原情况，通过Darboux变换方法求解了四分量反应扩散方程和四分量改进的Korteweg-de Vries（mKdV）方程。