Why Are Certain Nonlinear Partial Differential Equations (PDE) Both Widely Applicable and Integrable

摘要

Derivation of universal nonlinear evolution partial differential equations (PDEs) by a limiting procedure involving rescalings and an asymptotic expansion, from very large classes of nonlinear evolution equations is considered. Because this limiting procedure reveals 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 discussed. The heuristic distinction is mentioned among equations that are linearizable by an appropriate change of variables, and equations that are integrable via the spectral transform technique.