Integrable Systems Of Hydrodynamic Type Research Articles

Integrable hierarchies, Frölicher–Nijenhuis bicomplexes and Lauricella bi-flat F-manifolds

Given the Frölicher-Nijenhuis bicomplex associated with a -tensor field L with vanishing Nijenhuis torsion, we define a multi-parameter family of bi-flat structures . This result is obtained by combining the construction of integrable hierarchies of hydrodynamic type starting from Frölicher-Nijenhuis bicomplexes with the construction of flat F-manifold structures from integrable systems of hydrodynamic type. By construction L is the operator of multiplication by the Euler vector field E and the number of parameters coincides with the number of Jordan blocks appearing in its Jordan normal form. We call these structures Lauricella bi-flat structures since in the n-dimensional semisimple case flat coordinates of are Lauricella functions. The -tensor fields defining the corresponding integrable hierarchies have a similar block diagonal structure.

On Algebraic-Geometric Methods for Constructing Submanifolds with Flat Normal Bundle and Holonomic Net of Curvature Lines

In this paper we propose a generalization of Krichever’s algebraic-geometric construction of orthogonal coordinate systems in a flat space. In the theory of integrable systems of hydrodynamic type a fundamental role is also played by orthogonal coordinates in some special nonflat spaces. The most important class of such spaces is given by metrics of submanifolds in flat spaces that have flat normal bundle and holonomic net of curvature lines, which defines orthogonal coordinates on the submanifold. We propose a method for constructing such submanifolds from algebraic-geometric data. Explicit examples are presented.

On the dispersionless Davey–Stewartson system: Hamiltonian vector field Lax pair and relevant nonlinear Riemann–Hilbert problem for dDS-II system

In this paper, the semiclassical limit of Davey-Stewartson systems are studied. It shows that these dispersionless limited integrable systems of hydrodynamic type, which are defined as dDS (dispersionless Davey-Stewartson) systems, are arisen from the commutation condition of Lax pairs of one-parameter vector fields. The relevant nonlinear Riemann-Hilbert problems with some symmetries for the dDS-II system are also constructed. This kind of Riemann-Hilbert problems are meaningful for applying the formal inverse scattering transform method, recently developed by Manakov and Santini, to study the dDS-II system .

Haantjes Manifolds with Symmetry

We introduce the definition of Haantjes manifolds with symmetry and explain why these manifolds appear in the theory of integrable systems of hydrodynamic type and in topological field theories.

An Effective Algorithm for Finding Multidimensional Conservation Laws for Integrable Systems of Hydrodynamic Type

We study a new property of integrable systems, the existence of infinitely many local three-dimensional conservation laws for pairs of integrable two-dimensional hydrodynamic chains. We describe an effective algorithm for successively computing an infinite set of three-dimensional conservation laws for the Benney pair of commuting flows.

Confluence of hypergeometric functions and integrable hydrodynamic-type systems

It is known that a large class of integrable hydrodynamic type systems can be constructed through the Lauricella function, a generalization of the classical Gauss hypergeometric function. In this paper, we construct novel class of integrable hydrodynamic type systems which govern the dynamics of critical points of confluent Lauricella type functions defined on finite dimensional Grassmannian Gr(2,n), the set of 2xn matrices of rank two. Those confluent functions satisfy certain degenerate Euler-Poisson-Darboux equations. It is also shown that in general, hydrodynamic type system associated to the confluent Lauricella function is given by an integrable and non-diagonalizable quasi-linear system of a Jordan matrix form. The cases of Grassmannian Gr(2,5) for two component systems and Gr(2,6) for three component systems are considered in details.

Quantum cohomology and quantum hydrodynamics from supersymmetric quiver gauge theories

We study the connection between N = 2 supersymmetric gauge theories, quantum cohomology and quantum integrable systems of hydrodynamic type. We consider gauge theories on ALE spaces of A and D -type and discuss how they describe the quantum cohomology of the corresponding Nakajima’s quiver varieties. We also discuss how the exact evaluation of local BPS observables in the gauge theory can be used to calculate the spectrum of quantum Hamiltonians of spin Calogero integrable systems and spin Intermediate Long Wave hydrodynamics. This is explicitly obtained by a Bethe Ansatz Equation provided by the quiver gauge theory in terms of its adjacency matrix.

Six-dimensional supersymmetric gauge theories, quantum cohomology of instanton moduli spaces and gl(N) Quantum Intermediate Long Wave Hydrodynamics

We show that the exact partition function of U(N) six-dimensional gauge theory with eight supercharges on C^2 x S^2 provides the quantization of the integrable system of hydrodynamic type known as gl(N) periodic Intermediate Long Wave (ILW). We characterize this system as the hydrodynamic limit of elliptic Calogero-Moser integrable system. We compute the Bethe equations from the effective gauged linear sigma model on S^2 with target space the ADHM instanton moduli space, whose mirror computes the Yang-Yang function of gl(N) ILW. The quantum Hamiltonians are given by the local chiral ring observables of the six-dimensional gauge theory. As particular cases, these provide the gl(N) Benjamin-Ono and Korteweg-de Vries quantum Hamiltonians. In the four dimensional limit, we identify the local chiral ring observables with the conserved charges of Heisenberg plus W_N algebrae, thus providing a gauge theoretical proof of AGT correspondence.

Integrable multi-phase thermodynamic systems and Tsallis' composition rule

We derive a class of equations of state for a multi-phase thermodynamic system associated with a finite set of order parameters that satisfy an integrable system of hydrodynamic type. As particular examples, we discuss one-phase systems such as the van der Waals gas and the effective molecular field model. The case of N–phase systems is also discussed in detail in connection with entropies depending on the order parameter according to Tsallis' composition rule.

Generalized hydrodynamic reductions of the kinetic equation for a soliton gas

We derive generalized multiflow hydrodynamic reductions of the nonlocal kinetic equation for a soliton gas and investigate their structure. These reductions not only provide further insight into the properties of the new kinetic equation but also could prove to be representatives of a novel class of integrable systems of hydrodynamic type beyond the conventional semi-Hamiltonian framework.

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.

Kinetic Equation for a Soliton Gas and Its Hydrodynamic Reductions

We introduce and study a new class of kinetic equations, which arise in the description of nonequilibrium macroscopic dynamics of soliton gases with elastic collisions between solitons. These equations represent nonlinear integro-differential systems and have a novel structure, which we investigate by studying in detail the class of N-component ‘cold-gas’ hydrodynamic reductions. We prove that these reductions represent integrable linearly degenerate hydrodynamic type systems for arbitrary N which is a strong evidence in favour of integrability of the full kinetic equation. We derive compact explicit representations for the Riemann invariants and characteristic velocities of the hydrodynamic reductions in terms of the ‘cold-gas’ component densities and construct a number of exact solutions having special properties (quasiperiodic, self-similar). Hydrodynamic symmetries are then derived and investigated. The obtained results shed light on the structure of a continuum limit for a large class of integrable systems of hydrodynamic type and are also relevant to the description of turbulent motion in conservative compressible flows.

A coordinate-free construction of conservation laws and reciprocal transformations for a class of integrable hydrodynamic-type systems

Using a (1, 1)-tensor L with zero Nijenhuis torsion and maximal possible number (equal to the number of dependent variables) of distinct, functionally independent eigenvalues we define, in a coordinate-free fashion, the seed systems which are weakly nonlinear semi-Hamiltonian systems of a special form, and an infinite set of conservation laws for the seed systems. The reciprocal transformations constructed from these conservation laws yield a considerably larger class of hydrodynamic-type systems from the seed systems, and we show that these new systems are again defined in a coordinate-free manner, using the tensor L alone, and, moreover, are weakly nonlinear and semi-Hamiltonian, so their general solution can be obtained by means of the generalized hodograph method of Tsarev.

Hydrodynamic type integrable equations on a segment and a half-line

The concept of integrable boundary conditions is applied to hydrodynamic type systems. Examples of such boundary conditions for dispersionless Toda systems are obtained. The close relation of integrable boundary conditions with integrable reductions in multifield systems is observed. The problem of consistency of boundary conditions with the Hamiltonian formulation is discussed. Examples of Hamiltonian integrable hydrodynamic type systems on a segment and a semiline are presented.

On (2+1)-dimensional hydrodynamic type systems possessing a pseudopotential with movable singularities

We consider a class of hydrodynamic type systems that have three independent and N ⩾ 2 dependent variables and possess a pseudopotential. It turns out that systems having a pseudopotential with movable singularities can be described by some functional equation. We find all solutions of this equation, which permits constructing interesting examples of integrable systems of hydrodynamic type for arbitrary N.

A family of (2+1)-dimensional hydrodynamic type systems possessing a pseudopotential

We construct a family of integrable hydrodynamic type systems with three independent and n ≥ 2 dependent variables in terms of solutions of a linear system of PDEs with rational coefficients. We choose the existence of a pseudopotential as a criterion of integrability. In the case n = 2 this family is a general solution of the classification problem for such systems. We also give an elliptic analog of this family in the case n > 2.

Algebro-Geometric Approach in the Theory of Integrable Hydrodynamic Type Systems

The algebro-geometric approach for integrability of semi-Hamiltonian hydrodynamic type systems is presented. This method is significantly simplified for so-called symmetric hydrodynamic type systems. Plenty interesting and physically motivated examples are investigated.

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.