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Why are certain nonlinear PDEs both widely applicable and integrable? (English) Zbl 0808.35001

What is integrability, Springer Ser. Nonlinear Dyn., 1-62 (1991).

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Summary: [For the entire collection see Zbl 0724.00014.]
Certain ‘universal’ nonlinear evolution PDEs can be obtained, by a limiting procedure involving rescalings and an asymptotic expansion, from very large classes of nonlinear evolution equations. Because this limiting procedure is the correct one to evince weakly nonlinear effects, these universal model equations show up in many applicative contexts. Because this limiting procedure generally preserves integrability, these universal model equations are likely to be integrable, since for this to happen it is sufficient that the very large class from which they are obtainable contains just one integrable equation.
The relevance and usefulness of this approach to understand the integrability of known equations, to test the integrability of new equations and to obtain novel integrable equations likely to be applicable are discussed concisely. In this context, the heuristic distinction is mentioned among ‘\(C\)-integrable’ and ‘\(S\)-integrable’ nonlinear PDEs, namely equations that are linearizable by an appropriate change of variables, and equations that are integrable via the spectral transform technique; several interesting \(C\)-integrable equations are reported.

MSC:

35-02 Research exposition (monographs, survey articles) pertaining to partial differential equations
35Q58 Other completely integrable PDE (MSC2000)
37J35 Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests
37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)

Citations:

Zbl 0724.00014
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