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Register a new userLax representations with non-removable parameters and integrable hierarchies of PDEs via exotic cohomology of symmetry algebras
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Authors
Oleg I. Morozov
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This paper develops the technique of constructing Lax representations for PDEs via non-central extensions generated by non-triivial exotic 2-cocycles of their contact symmetry algebras. We show that the method is applicable to the Lax representations with non-removable spectral parameters. Also we demonstrate that natural extensions of the symmetry algebras produce the integrable hierarchies associated to their PDEs.
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We consider the four-dimensional reduced quasi-classical self-dual Yang--Mills equation and show that non-triviality of the second exotic cohomology group of its symmetry algebra implies existence of a two-component integrable generalization of this equation. The sequence of natural extensions of this symmetry algebra generate an integrable hierarchy of multi-dimensional nonlinear PDEs. We write out the first three elements of this hierarchy.
In this paper we investigate integrable models from the perspective of information theory, exhibiting various connections. We begin by showing that compressible hydrodynamics for a one-dimesional isentropic fluid, with an appropriately motivated information theoretic extension, is described by a general nonlinear Schrodinger (NLS) equation. Depending on the choice of the enthalpy function, one obtains the cubic NLS or other modified NLS equations that have applications in various fields. Next, by considering the integrable hierarchy associated with the NLS model, we propose higher order information measures which include the Fisher measure as their first member. The lowest members of the hiearchy are shown to be included in the expansion of a regularized Kullback-Leibler measure while, on the other hand, a suitable combination of the NLS hierarchy leads to a Wootters type measure related to a NLS equation with a relativistic dispersion relation. Finally, through our approach, we are led to construct an integrable semi-relativistic NLS equation.
In this paper, we study explicit correspondences between the integrable Novikov and Sawada-Kotera hierarchies, and between the Degasperis-Procesi and Kaup-Kupershmidt hierarchies. We show how a pair of Liouville transformations between the isospectral problems of the Novikov and Sawada-Kotera equations, and the isospectral problems of the Degasperis-Procesi and Kaup-Kupershmidt equations relate the corresponding hierarchies, in both positive and negative directions, as well as their associated conservation laws. Combining these results with the Miura transformation relating the Sawada-Kotera and Kaup-Kupershmidt equations, we further construct an implicit relationship which associates the Novikov and Degasperis-Procesi equations.
A class of multidimensional integrable hierarchies connected with commutation of general (unreduced) (N+1)-dimensional vector fields containing derivative over spectral variable is considered. They are represented in the form of generating equation, as well as in the Lax-Sato form. A dressing scheme based on nonlinear vector Riemann problem is presented for this class. The hierarchies connected with Manakov-Santini equation and Dunajski system are considered as illustrative examples.
Paraconformal or $GL(2)$ geometry on an $n$-dimensional manifold $M$ is defined by a field of rational normal curves of degree $n-1$ in the projectivised cotangent bundle $mathbb{P} T^*M$. Such geometry is known to arise on solution spaces of ODEs with vanishing Wunschmann (Doubrov-Wilczynski) invariants. In this paper we discuss yet another natural source of $GL(2)$ structures, namely dispersionless integrable hierarchies of PDEs (for instance the dKP hierarchy). In the latter context, $GL(2)$ structures coincide with the characteristic variety (principal symbol) of the hierarchy. Dispersionless hierarchies provide explicit examples of various particularly interesting classes of $GL(2)$ structures studied in the literature. Thus, we obtain torsion-free $GL(2)$ structures of Bryant that appeared in the context of exotic holonomy in dimension four, as well as totally geodesic $GL(2)$ structures of Krynski. The latter, also known as involutive $GL(2)$ structures, possess a compatible affine connection (with torsion) and a two-parameter family of totally geodesic $alpha$-manifolds (coming from the dispersionless Lax equations), which makes them a natural generalisation of the Einstein-Weyl geometry. Our main result states that involutive $GL(2)$ structures are governed by a dispersionless integrable system. This establishes integrability of the system of Wunschmann conditions.
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