Method Of Hydrodynamic Reductions Research Articles

On a class of integrable Hamiltonian equations in 2+1 dimensions.

We classify integrable Hamiltonian equations of the form ut=∂x(δHδu),H=∫h(u,w) dxdy, where the Hamiltonian density h(u, w) is a function of two variables: dependent variable u and the non-locality . Based on the method of hydrodynamic reductions, the integrability conditions are derived (in the form of an involutive PDE system for the Hamiltonian density h). We show that the generic integrable density is expressed in terms of the Weierstrass σ-function: h(u, w) = σ(u) ew. Dispersionless Lax pairs, commuting flows and dispersive deformations of the resulting equations are also discussed.

Integrability, exact reductions and special solutions of the KP–Whitham equations

Reductions of the KP–Whitham system, namely the (2+1)-dimensional hydrodynamic system of five equations that describes the slow modulations of periodic solutions of the Kadomtsev–Petviashvili (KP) equation, are studied. Specifically, the soliton and harmonic wave limits of the KP–Whitham system are considered, which give rise in each case to a four-component (2+1)-dimensional hydrodynamic system. It is shown that a suitable change of dependent variables splits the resulting four-component systems into two parts: (i) a decoupled, independent two-component system comprised of the dispersionless KP equation, (ii) an auxiliary, two-component system coupled to the mean flow equations, which describes either the evolution of a linear wave or a soliton propagating on top of the mean flow. The integrability of both four-component systems is then demonstrated by applying the Haantjes tensor test as well as the method of hydrodynamic reductions. Various exact reductions of these systems are then presented that correspond to concrete physical scenarios.

On integrability in Grassmann geometries: integrable systems associated with fourfolds in Gr(3,5)

Let Gr(d; n) be the Grassmannian of d-dimensional linear subspaces of an n-dimensional vector space V n. A submanifold X Gr(d; n) gives rise to a differential system ⊂(X) that governs d-dimensional submanifolds of V n whose Gaussian image is contained in X. Systems of the form Σ(X) appear in numerous applications in continuum mechanics, theory of integrable systems, general relativity and differential geometry. They include such wellknown examples as the dispersionless Kadomtsev-Petviashvili equation, the Boyer-Finley equation, Plebansky's heavenly equations, and so on. In this paper we concentrate on the particularly interesting case of this construction where X is a fourfold in Gr(3; 5). Our main goal is to investigate differential-geometric and integrability aspects of the corresponding systems Σ(X). We demonstrate the equivalence of several approaches to dispersionless integrability such as • the method of hydrodynamic reductions, • the method of dispersionless Lax pairs, • integrability on solutions, based on the requirement that the characteristic variety of system Σ(X) defines an Einstein-Weyl geometry on every solution, • integrability on equation, meaning integrability (in twistor-theoretic sense) of the canonical GL(2;R) structure induced on a fourfold X ⊂ Gr(3; 5). All these seemingly different approaches lead to one and the same class of integrable systems Σ(X). We prove that the moduli space of such systems is 6-dimensional. We give a complete description of linearisable systems (the corresponding fourfold X is a linear section of Gr(3; 5)) and linearly degenerate systems (the corresponding fourfold X is the image of a quadratic map P4 99K Gr(3; 5)). The fourfolds corresponding to `generic' integrable systems are not algebraic, and can be parametrised by generalised hypergeometric functions.

Degenerate first-order Hamiltonian operators of hydrodynamic type in 2D

First-order Hamiltonian operators of hydrodynamic type were introduced by Drubrovin and Novikov in 1983. In 2D, they are generated by a pair of contravariant metrics g, and a pair of differential-geometric objects b, . If the determinant of the pencil vanishes for all λ, the operator is called degenerate. In this paper we provide a complete classification of degenerate two- and three-component Hamiltonian operators. Moreover, we study the integrability, by the method of hydrodynamic reductions, of 2+1 Hamiltonian systems arising from the structures we classified.

Dispersionless integrable systems in 3D and Einstein-Weyl geometry

For several classes of second-order dispersionless PDEs, we show that the symbols of their formal linearizations define conformal structures that must be Einstein-Weyl in 3D (or self-dual in 4D) if and only if the PDE is integrable by the method of hydrodynamic reductions. This demonstrates that the integrability of these dispersionless PDEs can be seen from the geometry of their formal linearizations.

On classification of integrable Davey–Stewartson type equations

This paper is devoted to the classification of integrable Davey–Stewartson type equations. A novel approach based on the requirement that such systems must be generated by a polynomial dispersionless Lax pair is used to obtain a list of potentially deformable dispersionless systems. Integrable dispersive systems and their Lax pairs are then constructed by employing a perturbative algorithm using the method of hydrodynamic reductions. Some of the found systems seem to be new.

Towards the classification of integrable differential–difference equations in 2 + 1 dimensions

We address the problem of classification of integrable differential–difference equations in 2 + 1 dimensions with one/two discrete variables. Our approach is based on the method of hydrodynamic reductions and its generalization to dispersive equations as proposed in Ferapontov et al (2009 J. Phys. A: Math. Theor. 42 035211, 2009 J. Phys. A: Math. Theor. 42 345205). We obtain a number of classification results of scalar integrable equations including that of the intermediate long wave and Toda type.

On the central quadric ansatz: integrable models and Painlevé reductions

It was observed by Tod (1995 Class. Quantum Grav.12 1535–47) and later by Dunajski and Tod (2002 Phys. Lett. A 303 253–64) that the Boyer–Finley (BF) and the dispersionless Kadomtsev–Petviashvili (dKP) equations possess solutions whose level surfaces are central quadrics in the space of independent variables (the so-called central quadric ansatz). It was demonstrated that generic solutions of this type are described by Painlevé equations PIII and PII, respectively. The aim of our paper is threefold: (1) Based on the method of hydrodynamic reductions, we classify integrable models possessing the central quadric ansatz. This leads to the five canonical forms (including BF and dKP). (2) Applying the central quadric ansatz to each of the five canonical forms, we obtain all Painlevé equations PI–PVI, with PVI corresponding to the generic case of our classification. (3) We argue that solutions coming from the central quadric ansatz constitute a subclass of two-phase solutions provided by the method of hydrodynamic reductions.

Classification of integrable two-component Hamiltonian systems of hydrodynamic type in 2 + 1 dimensions

Hamiltonian systems of hydrodynamic type occur in a wide range of applications including fluid dynamics, the Whitham averaging procedure, and the theory of Frobenius manifolds. In 1 + 1 dimensions, the requirement of the integrability of such systems by the generalised hodograph transform implies that integrable Hamiltonians depend on a certain number of arbitrary functions of two variables. On the contrary, in 2 + 1 dimensions the requirement of the integrability by the method of hydrodynamic reductions, which is a natural analogue of the generalised hodograph transform in higher dimensions, leads to finite-dimensional moduli spaces of integrable Hamiltonians. In this paper we classify integrable two-component Hamiltonian systems of hydrodynamic type for all existing classes of differential-geometric Poisson brackets in 2D, establishing a parametrisation of integrable Hamiltonians via elliptic/hypergeometric functions. Our approach is based on the Godunov-type representation of Hamiltonian systems, and utilises a novel construction of Godunov's systems in terms of generalised hypergeometric functions.

On Linear Degeneracy of Integrable Quasilinear Systems in Higher Dimensions

We investigate (d + 1)-dimensional quasilinear systems which are integrable by the method of hydrodynamic reductions. In the case d ≥ 3 we formulate a conjecture that any such system with an irreducible dispersion relation must be linearly degenerate. We prove this conjecture in the 2-component case, providing a complete classification of multi- dimensional integrable systems in question. In particular, our results imply the non- existence of 2-component integrable systems of hydrodynamic type for d ≥ 6. In the second half of the paper we discuss a numerical and analytical evidence for the impossibility of the breakdown of smooth initial data for linearly degenerate systems in 2 + 1 dimensions.

Classification of integrable hydrodynamic chains

Using the method of hydrodynamic reductions, we find all integrable infinite (1+1)-dimensional hydrodynamic-type chains of shift 1. A class of integrable infinite (2+1)-dimensional hydrodynamic-type chains is constructed.

SECOND-ORDER QUASILINEAR PDEs AND CONFORMAL STRUCTURES IN PROJECTIVE SPACE

We investigate second-order quasilinear equations of the form fijuxixj = 0, where u is a function of n independent variables x1, …, xn, and the coefficients fij depend on the first-order derivatives p1 = ux1, …, pn = uxn only. We demonstrate that the natural equivalence group of the problem is isomorphic to SL(n + 1, R), which acts by projective transformations on the space Pn with coordinates p1, …, pn. The coefficient matrix fij defines on Pn a conformal structure fij(p)dpidpj. The necessary and sufficient conditions for the integrability of such equations by the method of hydrodynamic reductions are derived, implying that the moduli space of integrable equations is 20-dimensional. Any equation satisfying the integrability conditions is necessarily conservative, and possesses a dispersionless Lax pair. The integrability conditions imply that the conformal structure fij(p) dpidpj is conformally flat, and possesses infinitely many three-conjugate null coordinate systems parametrized by three arbitrary functions of one variable. Integrable equations provide examples of such conformal structures parametrized by elementary functions, elliptic functions and modular forms.

Integrable Lagrangians and modular forms

We investigate non-degenerate Lagrangians of the form ∫ f ( u x , u y , u t ) d x d y d t such that the corresponding Euler–Lagrange equations ( f u x ) x + ( f u y ) y + ( f u t ) t = 0 are integrable by the method of hydrodynamic reductions. We demonstrate that the integrability conditions, which constitute an involutive over-determined system of fourth order PDEs for the Lagrangian density f , are invariant under a 20-parameter group of Lie-point symmetries whose action on the moduli space of integrable Lagrangians has an open orbit. The density of the ‘master-Lagrangian’ corresponding to this orbit is shown to be a modular form in three variables defined on a complex hyperbolic ball. We demonstrate how the knowledge of the symmetry group allows one to linearize the integrability conditions.

Integrable equations in 2 + 1 dimensions: deformations of dispersionless limits

We classify integrable third-order equations in 2 + 1 dimensions which generalize the examples of Kadomtsev–Petviashvili, Veselov–Novikov and Harry Dym equations. Our approach is based on the observation that dispersionless limits of integrable systems in 2 + 1 dimensions possess infinitely many multi-phase solutions coming from the so-called hydrodynamic reductions. In this paper, we adopt a novel perturbative approach to the classification problem. Based on the method of hydrodynamic reductions, we first classify integrable quasilinear systems which may (potentially) occur as dispersionless limits of soliton equations in 2 + 1 dimensions. To reconstruct dispersive deformations, we require that all hydrodynamic reductions of the dispersionless limit be inherited by the corresponding dispersive counterpart. This procedure leads to a complete list of integrable third-order equations, some of which are apparently new.

Classification of integrable hydrodynamic chains and generating functions of conservation laws

New approach to classification of integrable hydrodynamic chains is established. Generating functions of conservation laws are classified by the method of hydrodynamic reductions. N parametric family of explicit hydrodynamic reductions allows to reconstruct corresponding hydrodynamic chains. Plenty new hydrodynamic chains are found.

On a Class of Three-Dimensional Integrable Lagrangians

We characterize non-degenerate Lagrangians of the form such that the corresponding Euler-Lagrange equations are integrable by the method of hydrodynamic reductions. The integrability conditions constitute an over-determined system of fourth order PDEs for the Lagrangian density f, which is in involution and possesses interesting differential-geometric properties. The moduli space of integrable Lagrangians, factorized by the action of a natural equivalence group, is three-dimensional. Familiar examples include the dispersionless Kadomtsev-Petviashvili (dKP) and the Boyer-Finley Lagrangians, f=u x 3/3+u y 2−u x u t and , respectively. A complete description of integrable cubic and quartic Lagrangians is obtained. Up to the equivalence transformations, the list of integrable cubic Lagrangians reduces to three examples, There exists a unique integrable quartic Lagrangian, We conjecture that these examples exhaust the list of integrable polynomial Lagrangians which are essentially three-dimensional (it was verified that there exist no polynomial integrable Lagrangians of degree five). We prove that the Euler-Lagrange equations are integrable by the method of hydrodynamic reductions if and only if they possess a scalar pseudopotential playing the role of a dispersionless `Lax pair'.