The Coupled Modified Korteweg-de Vries Equations

Generalization of the modified KdV equation to a multi-component system, that is expressed by \(\frac{\partial u_i}{\partial t} + 6 \bigl( \sum_{j,k=0}^{M-1} C_{jk} u_j u_k \bigr) \frac{\partial u_i}{\partial x} + \frac{\partial^3 u_{i}}{\partial x^3} =0\), i =0, 1, …, M -1, is studied. We apply a new extended version of the inverse scattering method to this system. It is shown that this system has an infinite number of conservation laws and multi-soliton solutions. Further, the initial value problem of the model is solved.

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