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Almost integrable evolution equations. (English) Zbl 1022.35062

Summary: We present a 2-component equation with exactly two nontrivial generalized symmetries, a counterexample to Fokas’ conjecture that equations with as many symmetries as components are integrable. Furthermore we prove the existence of infinitely many evolution equations with finitely many symmetries. We introduce the concept of almost integrability to describe the situation where one has a finite number of symmetries. The symbolic calculus of Gel’fand-Dikii and \(p\)-adic analysis are used to prove our results.

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.)
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