Systems Of Hydrodynamic Type Research Articles

Multidimensional Nonhomogeneous Quasi-Linear Systems and Their Hamiltonian Structure

In this paper, we investigate multidimensional first-order quasi-linear systems and find necessary conditions for them to admit Hamiltonian formulation. The insufficiency of the conditions is related to the Poisson cohomology of the admissible Hamiltonian operators. We present in detail the examples of two-dimensional, two-components systems of hydrodynamic type and of a real reduction of the 3-waves system.

Nijenhuis geometry IV: conservation laws, symmetries and integration of certain non-diagonalisable systems of hydrodynamic type in quadratures

Abstract The paper contains two lines of results: the first one is a study of symmetries and conservation laws of gl -regular Nijenhuis operators. We prove the splitting theorem for symmetries and conservation laws of Nijenhuis operators, show that the space of symmetries of a gl -regular Nijenhuis operator forms a commutative algebra with respect to (pointwise) matrix multiplication. Moreover, all the elements of this algebra are strong symmetries of each other. We establish a natural relationship between symmetries and conservation laws of a gl -regular Nijenhuis operator and systems of the first and second companion coordinates. Moreover, we show that the space of conservation laws is naturally related to the space of symmetries in the sense that any conservation laws can be obtained from a single conservation law by multiplication with an appropriate symmetry. In particular, we provide an explicit description of all symmetries and conservation laws for gl -regular operators at algebraically generic points. The second line of results contains an application of the theoretical part to a certain system of partial differential equations of hydrodynamic type, which was previously studied by different authors, but mainly in the diagonalisable case. We show that this system is integrable in quadratures, i.e. its solutions can be found for almost all initial curves by integrating closed 1-forms and solving some systems of functional equations. The system is not diagonalisable in general, and construction and integration of such systems is an actively studied and explicitly stated problem in the literature.

Applications of Nijenhuis Geometry V: Geodesic Equivalence and Finite-Dimensional Reductions of Integrable Quasilinear Systems

We describe all metrics geodesically compatible with a gl\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extrm{gl}$$\\end{document}-regular Nijenhuis operator L. The set of such metrics is large enough so that a generic local curve γ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\gamma $$\\end{document} is a geodesic for a suitable metric g from this set. Next, we show that a certain evolutionary PDE system of hydrodynamic type constructed from L preserves the property of γ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\gamma $$\\end{document} to be a g-geodesic. This implies that every metric g geodesically compatible with L gives us a finite-dimensional reduction of this PDE system. We show that its restriction onto the set of g-geodesics is naturally equivalent to the Poisson action of Rn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {R}^n$$\\end{document} on the cotangent bundle generated by the integrals coming from geodesic compatibility.

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.

Darboux integrability for diagonal systems of hydrodynamic type

We prove that (1) diagonal systems of hydrodynamic type are Darboux integrable if and only if the corresponding systems for commuting flows are Darboux integrable, (2) systems for commuting flows are Darboux integrable if and only if the Laplace transformation sequences terminate, (3) Darboux integrable systems are necessarily semihamiltonian. We give geometric interpretation for Darboux integrability of such systems in terms of congruences of lines and in terms of solution orbits with respect to symmetry subalgebras, discuss known and new examples.

Algebraic-geometry approach to construction of semi-Hamiltonian systems of hydrodynamic type

In this paper, a class of semi-Hamiltonian diagonal systems of hydrodynamic type is constructed using algebraic-geometric methods. For such systems, hydrodynamic integrals and hydrodynamic symmetries are constructed from algebraic-geometric data. Besides, it is described what algebraic-geometric data distinguish in this class Hamiltonian diagonal systems with Hamiltonian structures defined by flat metrics (local Dubrovin-Novikov brackets) and metrics of constant curvature (nonlocal Mokhov-Ferapontov brackets).

Multiple Riemann wave solutions of the general form of quasilinear hyperbolic systems

The objective of this paper is to construct Riemann $k$-wave solutions of the general form of first-order quasilinear hyperbolic systems of partial differential equations geometrically. To this end, we adapt and combine elements of two approaches to the construction of Riemann $k$-waves, namely, the symmetry reduction method and the generalized method of characteristics. We formulate a geometrical setting for the general form of the $k$-wave problem and discuss in detail the conditions for the existence of $k$-wave solutions. An auxiliary result concerning the Frobenius theorem is established. We use it to obtain formulae describing the $k$-wave solutions in closed form. Our theoretical considerations are illustrated by examples of hydrodynamic type systems including the Brownian motion equation.

Kinetic Equation for Soliton Gas: Integrable Reductions

Macroscopic dynamics of soliton gases can be analytically described by the thermodynamic limit of the Whitham equations, yielding an integro-differential kinetic equation for the density of states. Under a delta-functional ansatz, the kinetic equation for soliton gas reduces to a non-diagonalisable system of hydrodynamic type whose matrix consists of several 2times 2 Jordan blocks. Here we demonstrate the integrability of this system by showing that it possesses a hierarchy of commuting hydrodynamic flows and can be solved by an extension of the generalised hodograph method. Our approach is a generalisation of Tsarev’s theory of diagonalisable systems of hydrodynamic type to quasilinear systems with non-trivial Jordan block structure.

Complete Integrability of a New Class of Hamiltonian Hydrodynamic Type Systems

In this paper we consider a new class of Hamiltonian hydrodynamic type systems, whose conservation laws are polynomial with respect to one of field variables.

Classification of bi-Hamiltonian pairs extended by isometries

The aim of this article is to classify pairs of the first-order Hamiltonian operators of Dubrovin–Novikov type such that one of them has a non-local part defined by an isometry of its leading coefficient. An example of such a bi-Hamiltonian pair was recently found for the constant astigmatism equation. We obtain a classification in the case of two dependent variables, and a significant new example with three dependent variables that is an extension of a hydrodynamic-type system obtained from a particular solution of the Witten–Dijkgraaf–Verlinde–Verlinde equations.

Poisson Brackets of Hydrodynamic Type and Their Generalizations

In this paper, we consider Hamiltonian structures of hydrodynamic type and some of their generalizations. In particular, we discuss the questions concerning the structure and special forms of the corresponding Poisson brackets and the connection of such structures with the theory of integration of systems of hydrodynamic type.

The Bifurcations and Exact Traveling Wave Solutions for a Nonlocal Hydrodynamic-Type System

A hydrodynamic-type system taking into account nonlocal effects is investigated. The exact traveling wave solutions including smooth solitary waves solutions, pseudo-peakons, periodic peakons, compactons, kink, and anti-kink wave solutions and so on are derived via the method of dynamical systems and the theory of singular traveling wave systems. It is worth pointing out that the uncountably infinitely many solitary wave solutions, kink, and anti-kink solutions are new solutions for the modeling system.

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.

Generalized symmetries, conservation laws and Hamiltonian structures of an isothermal no-slip drift flux model

We study the hydrodynamic-type system of differential equations modeling isothermal no-slip drift flux. Using the facts that the system is partially coupled and its subsystem reduces to the (1+1)-dimensional Klein–Gordon equation, we exhaustively describe generalized symmetries, cosymmetries and local conservation laws of this system. A generating set of local conservation laws under the action of generalized symmetries is proved to consist of two zeroth-order conservation laws. The subspace of translation-invariant conservation laws is singled out from the entire space of local conservation laws. We also find broad families of local recursion operators and a nonlocal recursion operator, and construct an infinite family of Hamiltonian structures involving an arbitrary function of a single argument. For each of the constructed Hamiltonian operators, we obtain the associated algebra of Hamiltonian symmetries.

Quasilinear systems of Jordan block type and the mKP hierarchy

Hydrodynamic type systems in Riemann invariants arise in a whole range of applications in fluid dynamics, Whitham averaging procedure, differential geometry and the theory of Frobenius manifolds. In this paper we discuss parabolic (Jordan block) analogues of diagonalisable systems. Our main observation is that integrable quasilinear systems of Jordan block type are parametrised by solutions of the modified Kadomtsev–Petviashvili hierarchy. Such systems appear naturally as degenerations of quasilinear systems associated with multi-dimensional hypergeometric functions, in the context of parabolic regularisation of the Riemann equation, as finite-component reductions of hydrodynamic chains, and as hydrodynamic reductions of linearly degenerate dispersionless integrable PDEs in multi-dimensions.

THE HIERARCHY OF ASSOCIATIVITY EQUATIONS FOR n=3 CASE WITH AN METRIC ƞ11≠0

This paper describes the hierarchy for N = 2 and n=3 case with an metric ƞ11≠0 when V0 = 0 of associativity equations. The equation of associativity arose from the 2D topological field theory. 2D topological field theory represent the matter sector of topological string theory. These theories covariant before coupling to gravity due to the presence of a nilpotent symmetry and are therefore often referred to as cohomological field theories. We give a description of nonlinear partial differential equations of associativity in 2D topological field theories as integrable nondiagonalizable weakly nonlinear homogeneous system of hydrodynamic type. The article discusses nonlinear equations of the third order for a function f = f(x,t)) of two independent variables x, t. In this work we consider the associativity equation for n=3 case with an a metric 0 11   . The solution of some cases of hierarchy when N = 2 and V0 = 0 equations of associativity illustrated

Dispersionless limit of the (1+1)-dimensional Fokas-Lenells equation

Nonlinear dispersionless equations can be obtained as dispersionless limits (quasiclassical limits) of integrable hierarchies of equations or by constructing a system of hydrodynamic type. In this paper, the dispersionless limit of the (1+1)-dimensional Fokas-Lenells equation is found. We present Lax pair for this dispersionless equation. As well known, the Fokas-Lenells equation characterizes the propagation of ultrashort nonlinear light pulses in optical fibers. Exact solutions of the dispersionless version of the Fokas-Lenells equation describe nonlinear shock waves in optical fiber systems. By using the obtained results, one can find shock wave solutions of the dispersionless equation, which have different physical applications.

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 .

ИНТЕГРИРУЕМЫЕ ПО ЛИУВИЛЛЮ РЕДУКЦИИ УРАВНЕНИЙ АССОЦИАТИВНОСТИ НА МНОЖЕСТВО СТАЦИОНАРНЫХ ТОЧЕК ИНТЕГРАЛА В СЛУЧАЕ ТРЕХ ПРИМАРНЫХ ПОЛЕЙ

In this work, in the case of three primary fields, a reduction of the associativity equations (the Witten–Dijkgraaf–Verlinde–Verlinde system, see (Witten, 1990, Dijkgraaf et al., 1991, Dubrovin, 1994) with antidiagonal matrix ηij on the set of stationary points of a nondegenerate integral quadratic with respect to the first-order partial derivatives is constructed in an explicit form and its Liouville integrability is proved. In Mokhov’s paper (Mokhov, 1995, Mokhov, 1998), these associativity equations were presented in the form of an integrable nondiagonalizable system of hydrodynamic type. In the papers (Ferapontov, Mokhov, 1996, Ferapontov et al., 1997, Mokhov, 1998), a bi-Hamiltonian representation for these equations and a nondegenerate integral quadratic with respect to the first-order partial derivatives were found. Using Mokhov’s construction on canonical Hamiltonian property of an arbitrary evolutionary system on the set of stationary points of its nondegenerate integral of the papers (Mokhov, 1984, Mokhov, 1987), we construct explicitly the reduction for the integral quadratic with respect to the first-order partial derivatives, found explicitly the Hamiltonian of the corresponding canonical Hamiltonian system. We also found three functionally-independent integrals in involution with respect to the canonical Poisson bracket on the phase space for the constructed reduction of the associativity equations and thus proved the Liouville integrability of this reduction. This work is supported by the Russian Science Foundation under grant No. 18-11-00316.

КВАЗИ-ЛАГРАНЖЕВОЕ ИНТЕГРИРОВАНИЕ ДВУМЕРНЫХ УРАВНЕНИЙ ПРАНДТЛЯ И БУССИНЕСКА В НЕВЯЗКОМ ПРЕДЕЛЕ

For the hydrodynamic type systems locally describing incompressible twodimensional fluid flows without dissipation we suggest quasi-Lagrangian approach for their integration. This method is based on application of incomplete Legendre transformation when the independent variables become a Lagrangian invariant (i.e. the constant quantity along the fluid particle trajectory), instead of one of the spatial coordinate, and the rest ones which are another spatial coordinate and time. Thus, this method is based on the inverse transform of one of the spatial coordinate and by this reason differs from the the complete Legendre transformation. The classical example of the complete Legendre transformation is the Hodograph transformation applying to solve the equations for one-dimensional isoentropycal gas flows. In this paper it is shown that equation for the Lagrangian invariant after applying the incomplete Legendre transformation and introducing the stream function transforms into the linear eqation can be resolved by means of the generating function introduction. This method is turned to be effective for solving the inviscid twodimensional Prandtl equation (this equation describes the boundary layer behavior) that allows one to integrate this equation completely. In the case of the constant pressure along the boundary the parallel velocity component represents the Lagrangian invariant. The obtained solution is written through the initial data and satisfies the non-penetrate boundary condition. Analysis of this solution shows the formation of the singularity for the velocity gradient on the wall. This singularity appears as the result of breaking. At the breaking point the velocity gradient tends to infinity according to the power ~(t0–t)-1 where t0 is the singular time. This solution describes the appearance of the folding type singularity. It is shown also that the Prandtl equation admits complete integration for arbitrary dependence of pressure on the longitudinal coordinate. The simplest solution is written for the case of the constant pressure gradient. For the Boussinesq system the equation for density can be resolved by this method that reduces to one equation for the generating function. This work was supported by the RAS Presidium Program «Nonlinear dynamics: fundamental problems and applications».