Nonlocal infinite dimensional symmetries and linearization maps

James Hornick (McMaster)

Monday, March 30, 11:30 (HH403/AIMS Lab)

We study the linearization of nonlinear partial differential equations through the structure of their infinite-dimensional symmetry algebras (IDS). Two distinct regimes arise: in the first, the IDS can be realized as a local symmetry (possibly in a covering), in which case it directly encodes both the target linear equation and the linearizing transformation; in the second, the IDS cannot be made local, and symmetry-derived nonlinear superposition principles (NLSPs)—which rely on a local IDS—fail. The mapping-equation approach remains applicable, but becomes nontrivial: the target linear equation is no longer known a priori, and the construction of the mapping requires nontrivial ansatz choices.

For the Kundu–Eckhaus equation, despite the fact that a single local conservation law gives rise to a covering which admits a linearization map, the IDS is not realized as a local symmetry in any conservation-law-generated covering. Despite this obstruction, linearizability can still be detected intrinsically. We show that the existence of a linearization map in a given covering implies the presence of an infinite family of local adjoint symmetries (IDAS) parameterized by solutions of the adjoint target equation, this provides a generalization of the classical IDS-based approach to detecting hidden linear structure and identifying candidate target equations.

We reduce the problems of detecting linearizability, identifying the target equation, and constructing the linearization map to the problem of solving the symmetry determining equations, which can be carried out using standard symmetry software. The method is applied to recover the Kundu–Eckhaus–Schrödinger mapping.