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Register a new userLoewner equations and reductions of dispersionless hierarchies
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The equations of Loewner type can be derived in two very different contexts: one of them is complex analysis and the theory of parametric conformal maps and the other one is the theory of integrable systems. In this paper we compare the both approaches. After recalling the derivation of Lowner equations based on complex analysis we review one- and multi-variable reductions of dispersionless integrable hierarhies (dKP, dBKP, dToda, and dDKP). The one-vaiable reductions are described by solutions of differe
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Interpretation of dispersionless integrable hierarchies as equations of coisotropic deformations for certain algebras and other algebraic structures like Jordan triple systInterpretation of dispersionless integrable hierarchies as equations of coisotropic deformations for certain algebras and other algebraic structures like Jordan triple systems is discussed. Several generalizations are considered. Stationary reductions of the dispersionless integrable equations are shown to be connected with the dynamical systems on the plane completely integrable on a fixed energy level. ems is discussed. Several generalizations are considered. Stationary reductions of the dispersionless integrable equations are shown to be connected with the dynamical systems on the plane completely integrable on a fixed energy level.
The paper begins with a review of the well known Novikovs equations and corresponding finite KdV hierarchies. For a positive integer $N$ we give an explicit description of the $N$-th Novikovs equation and its first integrals. Its finite KdV hierarchy consists of $N$ compatible integrable polynomial dynamical systems in $mathbb{C}^{2N}$. Then we discuss a non-commutative version of the $N$-th Novikovs equation defined on a finitely generated free associative algebra $mathfrak{B}_N$ with $2N$ generators. In $mathfrak{B}_N$, for $N=1,2,3,4$, we have found two-sided homogeneous ideals $mathfrak{Q}_Nsubsetmathfrak{B}_N$ (quantisation ideals) which are invariant with respect to the $N$-th Novikovs equation and such that the quotient algebra $mathfrak{C}_N = mathfrak{B}_Ndiagup mathfrak{Q}_N$ has a well defined Poincare-Birkhoff-Witt basis. It enables us to define the quantum $N$-th Novikovs equation on the $mathfrak{C}_N$. We have shown that the quantum $N$-th Novikovs equation and its finite hierarchy can be written in the standard Heisenberg form.
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.
We prove that integrability of a dispersionless Hirota type equation implies the symplectic Monge-Ampere property in any dimension $geq 4$. In 4D this yields a complete classification of integrable dispersionless PDEs of Hirota type through a list of heavenly type equations arising in self-dual gravity. As a by-product of our approach we derive an involutive system of relations characterising symplectic Monge-Ampere equations in any dimension. Moreover, we demonstrate that in 4D the requirement of integrability is equivalent to self-duality of the conformal structure defined by the characteristic variety of the equation on every solution, which is in turn equivalent to the existence of a dispersionless Lax pair. We also give a criterion of linerisability of a Hirota type equation via flatness of the corresponding conformal structure, and study symmetry properties of integrable equations.
This is a short review of the Kadomtsev-Petviashvili hierarchies of types B and C. The main objects are the $L$-operator, the wave operator, the auxiliary linear problems for the wave function, the bilinear identity for the wave function and the tau-function. All of them are discussed in the paper. The connections with the usual Kadomtsev-Petviashvili hierarchy (of the type A) are clarified. Examples of soliton solutions and the dispersionless limit of the hierarchies are also considered.
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