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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.

35Q53KdV-like (Korteweg-de Vries) equations
37K10Completely integrable systems, integrability tests, bi-Hamiltonian structures, hierarchies
37K40Soliton theory, asymptotic behavior of solutions
39A12Discrete version of topics in analysis
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