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Integrable semi-discretization of the coupled modified KdV equations. (English) Zbl 0933.35176

Summary: The discrete version of the inverse scattering method proposed by Ablowitz and Ladik is extended to solve multi-component systems \[ {\partial u^{(i)}_n\over \partial t}=\left(1+ \sum^{M-1}_{j,k=0} C_{jk} u_n^{(j)} u_n^{(k)}\right) \bigl(u^{(i)}_{u+1} -u_{n-1}^{(i)}\bigr), \quad i=0, 1,\dots, M-1. \] The extension enables one to solve the initial value problem, which proves directly the complete integrability of a semi-discrete version of the coupled modified Korteweg-de Vries equations and their hierarchy. It also provides a procedure to obtain conservation laws and multi-soliton solutions of the hierarchy.

MSC:

35Q53 KdV equations (Korteweg-de Vries equations)
37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
37K40 Soliton theory, asymptotic behavior of solutions of infinite-dimensional Hamiltonian systems
39A12 Discrete version of topics in analysis

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