9,921 research outputs found
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.Comment: Latex, 34 page
Conditional symmetries and Riemann invariants for inhomogeneous hydrodynamic-type systems
A new approach to the solution of quasilinear nonelliptic first-order systems
of inhomogeneous PDEs in many dimensions is presented. It is based on a version
of the conditional symmetry and Riemann invariant methods. We discuss in detail
the necessary and sufficient conditions for the existence of rank-2 and rank-3
solutions expressible in terms of Riemann invariants. We perform the analysis
using the Cayley-Hamilton theorem for a certain algebraic system associated
with the initial system. The problem of finding such solutions has been reduced
to expanding a set of trace conditions on wave vectors and their profiles which
are expressible in terms of Riemann invariants. A couple of theorems useful for
the construction of such solutions are given. These theoretical considerations
are illustrated by the example of inhomogeneous equations of fluid dynamics
which describe motion of an ideal fluid subjected to gravitational and Coriolis
forces. Several new rank-2 solutions are obtained. Some physical interpretation
of these results is given.Comment: 19 page
Method of group foliation, hodograph transformation and non-invariant solutions of the Boyer-Finley equation
We present the method of group foliation for constructing non-invariant
solutions of partial differential equations on an important example of the
Boyer-Finley equation from the theory of gravitational instantons. We show that
the commutativity constraint for a pair of invariant differential operators
leads to a set of its non-invariant solutions. In the second part of the paper
we demonstrate how the hodograph transformation of the ultra-hyperbolic version
of Boyer-Finley equation in an obvious way leads to its non-invariant solution
obtained recently by Manas and Alonso. Due to extra symmetries, this solution
is conditionally invariant, unlike non-invariant solutions obtained previously.
We make the hodograph transformation of the group foliation structure and
derive three invariant relations valid for the hodograph solution, additional
to resolving equations, in an attempt to obtain the orbit of this solution.Comment: to appear in the special issue of Theor. Math. Phys. for the
Proceedings of NEEDS2002; Keywords: Heavenly equation, group foliation,
non-invariant solutions, hodograph transformatio
Integrable equations of the dispersionless Hirota type and hypersurfaces in the Lagrangian Grassmannian
We investigate integrable second order equations of the form
F(u_{xx}, u_{xy}, u_{yy}, u_{xt}, u_{yt}, u_{tt})=0.
Familiar examples include the Boyer-Finley equation, the potential form of
the dispersionless Kadomtsev-Petviashvili equation, the dispersionless Hirota
equation, etc. The integrability is understood as the existence of infinitely
many hydrodynamic reductions. We demonstrate that the natural equivalence group
of the problem is isomorphic to Sp(6), revealing a remarkable correspondence
between differential equations of the above type and hypersurfaces of the
Lagrangian Grassmannian. We prove that the moduli space of integrable equations
of the dispersionless Hirota type is 21-dimensional, and the action of the
equivalence group Sp(6) on the moduli space has an open orbit.Comment: 32 page
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