460 research outputs 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 ,
which is in involution and possess 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, 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'. MSC: 35Q58, 37K05, 37K10, 37K25. Keywords:
Multi-dimensional Dispersionless Integrable Systems, Hydrodynamic Reductions,
Pseudopotentials.Comment: 12 pages A4 format, standard Latex 2e. In the file progs.tar we
include the programs needed for computations performed in the paper. Read
1-README first. The new version includes two new section
Multiple Factorizations of Bivariate Linear Partial Differential Operators
We study the case when a bivariate Linear Partial Differential Operator
(LPDO) of orders three or four has several different factorizations.
We prove that a third-order bivariate LPDO has a first-order left and right
factors such that their symbols are co-prime if and only if the operator has a
factorization into three factors, the left one of which is exactly the initial
left factor and the right one is exactly the initial right factor. We show that
the condition that the symbols of the initial left and right factors are
co-prime is essential, and that the analogous statement "as it is" is not true
for LPDOs of order four.
Then we consider completely reducible LPDOs, which are defined as an
intersection of principal ideals. Such operators may also be required to have
several different factorizations. Considering all possible cases, we ruled out
some of them from the consideration due to the first result of the paper. The
explicit formulae for the sufficient conditions for the complete reducibility
of an LPDO were found also
Staeckel systems generating coupled KdV hierarchies and their finite-gap and rational solutions
We show how to generate coupled KdV hierarchies from Staeckel separable
systems of Benenti type. We further show that solutions of these Staeckel
systems generate a large class of finite-gap and rational solutions of cKdV
hierarchies. Most of these solutions are new.Comment: 15 page
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
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