On Integrable Systems & Rigidity for PDEs with Symmetry (2024)

Il Nuovo Cimento B (1971- …, 1984

Japanese Journal of Mathematics, 2013

arXiv (Cornell University), 2017

International Journal of Geometric Methods in Modern Physics, 2013

We review the theory of nonlocal symmetries of nonlinear partial differential equations and, as examples, we present infinite-dimensional Lie algebras of nonlocal symmetries of the Fokas–Qiao and Kaup–Kupershmidt equations. Then, we consider nonlocal symmetries of a family which contains the Korteweg–de Vries (KdV) and (a subclass of) the Rosenau–Hyman compacton-bearing K(m, n) equations. We find that the only member of the family which possesses nonlocal symmetries (of a kind specified in Sec. 3 below) is precisely the KdV equation. We take this fact as an indication that the K(m, n) equations are not integrable in general, and we use the formal symmetry approach of Shabat to check this claim: we prove that the only integrable cases of the full K(m, n) family are the KdV and modified KdV equations.

Journal of Hyperbolic Differential Equations, 2009

International Mathematics Research Notices

Journal of Physics A: Mathematical and Theoretical

Communications in Contemporary Mathematics, 2021

Let [Formula: see text] be an [Formula: see text]-dimensional hom*ogeneous manifold and [Formula: see text] be the manifold of [Formula: see text]-jets of hypersurfaces of [Formula: see text]. The Lie group [Formula: see text] acts naturally on each [Formula: see text]. A [Formula: see text]-invariant partial differential equation of order [Formula: see text] for hypersurfaces of [Formula: see text] (i.e., with [Formula: see text] independent variables and [Formula: see text] dependent one) is defined as a [Formula: see text]-invariant hypersurface [Formula: see text]. We describe a general method for constructing such invariant partial differential equations for [Formula: see text]. The problem reduces to the description of hypersurfaces, in a certain vector space, which are invariant with respect to the linear action of the stability subgroup [Formula: see text] of the [Formula: see text]-prolonged action of [Formula: see text]. We apply this approach to describe invariant partial ...

Mathematics

Functional Analysis and Its Applications, 1996

Journal of Dynamical and Control Systems - J DYN CONTROL SYST, 2002

The Harry-Dym equation comes from geodesic flows on diffeomorphism groups. This fact has been observed before by the Marsden school. In this paper we show that the supersymmetric Harry-Dym equation arises from the geodesic flow on the superconformal group. We also show that the stabilizer of a point in the coadjoint representation of the Virasoro algebra endowed with a Sobolev norm consists of a space of projective vector fields. We also show that for each projective vector field, there exists a quadratic that satisfies a Neumann system.

Journal of Geometry and Physics, 2006

In this paper, after some recalls about Poisson cohom*ology, we first study what the general method is in order to obtain a bi-Hamiltonian formulation of a given Hamiltonian system by means of a deformation. Then we show that the bi-Hamiltonian formulation which results from the deformation of a Poisson structure by means of a suitable non-Noether symmetry cannot explain the complete integrability for a large class of Arnold–Liouville integrable systems; next we prove that the deformation must be made in this context by a suitable mastersymmetry. At last, we give several examples.

Theoretical and Mathematical Physics, 2007

Arxiv preprint math/0301212, 2003

Proceedings of Symposia in Pure Mathematics, 1970