Why Are Certain Nonlinear PDEs Both Widely Applicable and Integrable?

Abstract

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 contain 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, is tersely discussed. 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; and several interesting C-integrable equations are reported.