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Integrable couplings of soliton equations by perturbations. I: A general theory and application to the KdV hierarchy. (English) Zbl 1001.37061
The author discusses a problem of integrable couplings. That is, having an integrable system of evolution equations \(u_t=K(u)\), how to obtain a bigger (nontrivial) integrable system consisting of \(u_t=K(u)\) and \(v_t = S(u,v)\). “Using various perturbations around solutions of perturbed soliton equations being analytic with respect to a small perturbation parameter” the author developes “a theory for constructing integrable couplings of soliton equations”. Application of the theory to the KdV hierarchy is given.

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
35Q53 KdV equations (Korteweg-de Vries equations)
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