Systems Of Hydrodynamic Type Research Articles

Bihamiltonian Systems of Hydrodynamic Type and Reciprocal Transformations

We prove that under certain linear reciprocal transformation, an evolutionary PDE of hydrodynamic type that admits a bihamiltonian structure is transformed to a system of the same type which is still bihamiltonian

Riemann Invariants and Rank-k Solutions of Hyperbolic Systems

In this paper we employ a “direct method” to construct rank-k solutions, expressible in Riemann invariants, to hyperbolic system of first order quasilinear di!erential equations in many dimensions. The most important feature of our approach is the analysis of group invariance properties of these solutions and applying the conditional symmetry reduction technique to the initial equations. We discuss in detail the necessary and su"cient conditions for existence of these type of solutions. We demonstrate our approach through several examples of hydrodynamic type systems; new classes of solutions are obtained in a closed form.

Integrable quasiclassical deformations of cubic curves

A general scheme for determining and studying hydrodynamic type systems describing integrable deformations of algebraic curves is applied to cubic curves. Lagrange resolvents of the theory of cubic equations are used to derive and characterize these deformations.

The description of pairs of compatible first-order differential geometric poisson brackets

We show that bi-Hamiltonian structures of systems of hydrodynamic type can be described in terms of solutions of nonlinear equations. These equations can be integrated by the inverse scattering transform for arbitrary metrics as well as for flat ones. In particular, if one metric is flat and the other has a constant curvature, then the corresponding integrable system is a reduction of the Cherednik chiral-field model.

S-functions, reductions and hodograph solutions of the rth dispersionless modified KP and Dym hierarchies

We introduce an S-function formulation for the recently found r-th dispersionless modified KP and r-th dispersionless Dym hierarchies, giving also a connection of these $S$-functions with the Orlov functions of the hierarchies. Then, we discuss a reduction scheme for the hierarchies that together with the $S$-function formulation leads to hodograph systems for the associated solutions. We consider also the connection of these reductions with those of the dispersionless KP hierarchy and with hydrodynamic type systems. In particular, for the 1-component and 2-component reduction we derive, for both hierarchies, ample sets of examples of explicit solutions.

On Coercive Estimates for Solutions of Linear Systems of Hydrodynamic Type

Local and nonlocal coercive estimates in Lebesgue, Holder, and Morrey–Campanato spaces are proved for solutions of linear stationary systems of hydrodynamic type. For this purpose, the technique proposed by Italian mathematicians for the study of solutions to equations and systems of elliptic and parabolic types is developed. Bibliography: 14 titles.

Symmetry constraints for dispersionless integrable equations and systems of hydrodynamic type

Symmetry constraints for ( 2 + 1 ) -dimensional dispersionless integrable equations are considered. It is demonstrated that they naturally lead to certain systems of hydrodynamic type which arise within the reduction method. One also easily obtains an associated complex curve (Sato function) and corresponding generating equations. Dispersionless KP and 2DTL hierarchy are considered as illustrative examples.

A Simple Way of Making a Hamiltonian System Into a Bi-Hamiltonian One

Given a Poisson structure (or, equivalently, a Hamiltonian operator) $P$, we show that its Lie derivative $L_{\tau}(P)$ along a vector field $\tau$ defines another Poisson structure, which is automatically compatible with $P$, if and only if $[L_{\tau}^2(P),P]=0$, where $[\cdot,\cdot]$ is the Schouten bracket. We further prove that if $\dim\ker P\leq 1$ and $P$ is of locally constant rank, then all Poisson structures compatible with a given Poisson structure $P$ on a finite-dimensional manifold $M$ are locally of the form $L_{\tau}(P)$, where $\tau$ is a local vector field such that $L_{\tau}^2(P)=L_{\tilde\tau}(P)$ for some other local vector field $\tilde\tau$. This leads to a remarkably simple construction of bi-Hamiltonian dynamical systems. We also present a generalization of these results to the infinite-dimensional case. In particular, we provide a new description for pencils of compatible local Hamiltonian operators of Dubrovin--Novikov type and associated bi-Hamiltonian systems of hydrodynamic type. Key words: compatible Poisson structures, Hamiltonian operators, bi-Hamiltonian systems (= bihamiltonian systems), integrability, Schouten bracket, master symmetry, Lichnerowicz--Poisson cohomology, hydrodynamic type systems. MSC 2000: Primary: 37K10; Secondary: 37K05, 37J35

Integrable quasiclassical deformations of algebraic curves

A general scheme for determining and studying integrable deformations of algebraic curves is presented. The method is illustrated with the analysis of the hyperelliptic case. An associated multi-Hamiltonian hierarchy of systems of hydrodynamic type is characterized.

Symmetry constraints for real dispersionless Veselov-Novikov equation

Symmetry constraints for dispersionless integrable equations are discussed. It is shown that under symmetry constraints, the dispersionless Veselov-Novikov equation is reduced to the (1+1)-dimensional hydrodynamic-type systems.

A class of Einstein–Weyl spaces associated to an integrable system of hydrodynamic type

HyperCR Einstein–Weyl equations in 2+1 dimensions reduce to a pair of quasi-linear PDEs of hydrodynamic type. All solutions to this hydrodynamic system can in principle be constructed from a twistor correspondence, thus establishing the integrability. Simple examples of solutions including the hydrodynamic reductions yield new Einstein–Weyl structures.

The characterization of two-component (2+1)-dimensional integrable systems of hydrodynamic type

We obtain the necessary and sufficient conditions for a two-component (2+1)-dimensional system of hydrodynamic type to possess infinitely many hydrodynamic reductions. These conditions are in involution, implying that the systems in question are locally parametrized by 15 arbitrary constants. It is proved that all such systems possess three conservation laws of hydrodynamic type and, therefore, are symmetrizable in Godunov's sense. Moreover, all such systems are proved to possess a scalar pseudopotential which plays the role of the `dispersionless Lax pair'. We demonstrate that the class of two-component systems possessing a scalar pseudopotential is in fact identical with the class of systems possessing infinitely many hydrodynamic reductions, thus establishing the equivalence of the two possible definitions of the integrability. Explicit linearly degenerate examples are constructed.

Integrable hydrodynamic chains

A new approach for derivation of Benney-type moment chains and integrable hydrodynamic type systems is presented. New integrable hydrodynamic chains are constructed; all their hydrodynamical reductions are described and integrated. New (2+1) integrable hydrodynamic type systems are found.

Hydrodynamic reductions of the heavenly equation

We demonstrate that Plebański's first heavenly equation decouples in infinitely many ways into a triple of commuting (1 + 1)-dimensional systems of hydrodynamic type which satisfy the Egorov property. Solving these systems by the generalized hodograph method, one can construct exact solutions of the heavenly equation parametrized by arbitrary functions of one variable. We discuss explicit examples of hydrodynamic reductions associated with the equations of one-dimensional nonlinear elasticity, linearly degenerate systems and the equations of associativity.

The Liouville Canonical Form for Compatible Nonlocal Poisson Brackets of Hydrodynamic Type and Integrable Hierarchies

We reduce an arbitrary pair of compatible nonlocal Poisson brackets of hydrodynamic type generated by metrics of constant Riemannian curvature (compatible Mokhov–Ferapontov brackets) to a canonical form, find an integrable system describing all such pairs, and, for an arbitrary solution of this integrable system, i.e., for any pair of compatible Poisson brackets in question, construct (in closed form) integrable bi-Hamiltonian systems of hydrodynamic type possessing this pair of compatible Poisson brackets of hydrodynamic type. The corresponding special canonical forms of metrics of constant Riemannian curvature are considered. A theory of special Liouville coordinates for Poisson brackets is developed. We prove that the classification of these compatible Poisson brackets is equivalent to the classification of special Liouville coordinates for Mokhov–Ferapontov brackets.

On the central extensions of Poisson brackets of hydrodynamic type

Poisson brackets of hydrodynamic type were introduced to construct a theory of conservation system of hydrodynamic type. They lead to the algebraic and differentially geometrical theory of local translationally invariant Lie algebras of first order, Frobenius-type algebras and Novikov algebras. One of the main ways to construct new examples of these algebras and their corresponding Poisson brackets is through the central extensions. In Balinskii A A and Novikov S P (1985 Sov. Math. Dokl. 32 228–31), there is a general theory of the central extensions. In this paper, we give a further detailed study of two (non-trivial) types of Novikov algebras which directly decide an important kind of central extension of the Poisson brackets. We find that the Novikov algebras in the case τ = 3 are in the case τ = 0 and these two cases coincide in dimension ≤ 3. Moreover, they also coincide for the transitive cases in dimension 4.

Tri-Hamiltonian Structures of Egorov Systems of Hydrodynamic Type ∗

We prove a simple condition under which the metric corresponding to a diagonalizable semi-Hamiltonian hydrodynamic type system belongs to the class of Egorov (potential) metrics. For Egorov diagonal hydrodynamic type systems satisfying natural semisimplicity and homogeneity conditions, we prove necessary and sufficient conditions under which the third structure is local or corresponds to a metric of constant curvature. The results are illustrated by some well-known physical examples of such systems.

Formation of Nonlinear Structures in Hydrodynamic-Type Systems

The nonlinear dynamics of one-dimensional systems of hydrodynamic type is examined based on the Lagrangian description. The formation of essentially nonlinear, time-dependent structures within a compressible advection problem is considered. The conditions leading to formation of new regular as well as singular collapse-type structures and wavelet structures in the density profile are derived.

The anomalous diffusion of wave disturbances in hydrodynamic-type systems

It is shown that solutions of many model non-linear equations of the hydrodynamic type (including the Navier-Stokes equations in the theory of a viscous fluid) in the form of localized wave disturbances undergo anomalous decay and the diffusion-type spreading under the action of Gaussian additive force noise and multiplicative parametric noise that depend randomly on time. The universality of this anomaly (i.e. its independence of the specific form of the equations and, in many respects, of the characteristic properties of the noise) is demonstrated; this reflects the generalized Galilean invariance of the hydrodynamic-type equations and the Gaussian property of random sources. The integrability of certain important hydrodynamic models enables the general conclusions to be supported by particular analytic results. Examples of expressions for the mean flow velocities, developed under the action of random forces on algebraic solitons of the completely integrable equations: equations for internal waves (the Benjamin-Ono equations), the two-dimensional Korteweg-de Vries equation (the Petviashvili-Kadomtseve equation), and the two-dimensional non-linear Schrödinger equation (the Davey-Stewartson equations), are presented in terms of the error function. It is established that for stochastic equations of the hydrodynamic type rapidly decaying multiplicative noise leads to normal Fick diffusion only when there is no additive force noise. Force noise produces anomalous diffusion of wave disturbances, which is described by Richardson's law (with a cubic relation between the temporal and squared spatial scales). The result obtained confirms the universal nature of Richardson's law, which has been demonstrated previously for many examples of turbulent and wave stochastic processes [1, 2].

Invariant description of solutions of hydrodynamic-type systems in hodograph space: hydrodynamic surfaces

Hydrodynamic surfaces are solutions of hydrodynamic type systems viewed as non-parametrized submanifolds of the hodograph space. We propose an invariant differential-geometric characterization of hydrodynamic surfaces by expressing the curvature form of the characteristic web in terms of the reciprocal invariants.