63 research outputs found
On Lie algebras responsible for zero-curvature representations of multicomponent (1+1)-dimensional evolution PDEs
Zero-curvature representations (ZCRs) are one of the main tools in the theory
of integrable -dimensional PDEs. According to the preprint
arXiv:1212.2199, for any given -dimensional evolution PDE one can define
a sequence of Lie algebras , , such that representations
of these algebras classify all ZCRs of the PDE up to local gauge equivalence.
ZCRs depending on derivatives of arbitrary finite order are allowed.
Furthermore, these algebras provide necessary conditions for existence of
Backlund transformations between two given PDEs. The algebras are defined
in arXiv:1212.2199 in terms of generators and relations.
In the present paper, we describe some methods to study the structure of the
algebras for multicomponent -dimensional evolution PDEs. Using
these methods, we compute the explicit structure (up to non-essential nilpotent
ideals) of the Lie algebras for the Landau-Lifshitz, nonlinear
Schrodinger equations, and for the -component Landau-Lifshitz system of
Golubchik and Sokolov for any . In particular, this means that for the
-component Landau-Lifshitz system we classify all ZCRs (depending on
derivatives of arbitrary finite order), up to local gauge equivalence and up to
killing nilpotent ideals in the corresponding Lie algebras.
The presented methods to classify ZCRs can be applied also to other
-dimensional evolution PDEs. Furthermore, the obtained results can be
used for proving non-existence of Backlund transformations between some PDEs,
which will be described in forthcoming publications.Comment: 56 pages. arXiv admin note: text overlap with arXiv:1303.357
(Super-)integrable systems associated to 2-dimensional projective connections with one projective symmetry
Projective connections arise from equivalence classes of affine connections
under the reparametrization of geodesics. They may also be viewed as quotient
systems of the classical geodesic equation. After studying the link between
integrals of the (classical) geodesic flow and its associated projective
connection, we turn our attention to 2-dimensional metrics that admit one
projective vector field, i.e. whose local flow sends unparametrized geodesics
into unparametrized geodesics. We review and discuss the classification of
these metrics, introducing special coordinates on the linear space of solutions
to a certain system of partial differential equations, from which such metrics
are obtained. Particularly, we discuss those that give rise to free
second-order superintegrable Hamiltonian systems, i.e. which admit 2
additional, functionally independent quadratic integrals. We prove that these
systems are parametrized by the 2-sphere, except for 6 exceptional points where
the projective symmetry becomes homothetic.Comment: 23 pages, 5 table
Geometric aspects of higher order variational principles on submanifolds
The geometry of jets of submanifolds is studied, with special interest in the
relationship with the calculus of variations. A new intrinsic geometric
formulation of the variational problem on jets of submanifolds is given.
Working examples are provided.Comment: 17 page
Normal forms of two-dimensional metrics admitting exactly one essential projective vector field
We give a complete list of mutually non-diffeomorphic normal forms for the
two-dimensional metrics that admit one essential (i.e., non-homothetic)
projective vector field. This revises a result from the literature and extends
the results of two papers, by R.L. Bryant & G. Manno & V.S. Matveev (2008) and
V.S. Matveev (2012) respectively, solving a problem posed by Sophus Lie in
1882.Comment: 47 pages, 4 figure
Lie algebras responsible for zero-curvature representations of scalar evolution equations
Zero-curvature representations (ZCRs) are one of the main tools in the theory
of integrable PDEs. In particular, Lax pairs for (1+1)-dimensional PDEs can be
interpreted as ZCRs.
For any (1+1)-dimensional scalar evolution equation , we define a family
of Lie algebras which are responsible for all ZCRs of in the
following sense. Representations of the algebras classify all ZCRs of
the equation up to local gauge transformations. To achieve this, we find a
normal form for ZCRs with respect to the action of the group of local gauge
transformations.
As we show in other publications, using these algebras, one obtains some
necessary conditions for integrability of the considered PDEs (where
integrability is understood in the sense of soliton theory) and necessary
conditions for existence of a B\"acklund transformation between two given
equations. Examples of proving non-integrability and applications to obtaining
non-existence results for B\"acklund transformations are presented in other
publications as well.
In our approach, ZCRs may depend on partial derivatives of arbitrary order,
which may be higher than the order of the equation . The algebras
generalize Wahlquist-Estabrook prolongation algebras, which are responsible for
a much smaller class of ZCRs.
In this paper we describe general properties of and present generators
and relations for these algebras. In other publications we study the structure
of for equations of KdV, Krichever-Novikov, Kaup-Kupershmidt,
Sawada-Kotera types. Among the obtained algebras, one finds
infinite-dimensional Lie algebras of certain matrix-valued functions on
rational and elliptic algebraic curves.Comment: 23 pages; v4: some results have been moved to other preprint
On Lie algebras responsible for integrability of (1+1)-dimensional scalar evolution PDEs
Zero-curvature representations (ZCRs) are one of the main tools in the theory
of integrable PDEs. In particular, Lax pairs for (1+1)-dimensional PDEs can be
interpreted as ZCRs. In [arXiv:1303.3575], for any (1+1)-dimensional scalar
evolution equation , we defined a family of Lie algebras which are
responsible for all ZCRs of in the following sense. Representations of the
algebras classify all ZCRs of the equation up to local gauge
transformations. In [arXiv:1804.04652] we showed that, using these algebras,
one obtains necessary conditions for existence of a B\"acklund transformation
between two given equations. The algebras are defined in terms of
generators and relations. In this paper we show that, using the algebras
, one obtains some necessary conditions for integrability of
(1+1)-dimensional scalar evolution PDEs, where integrability is understood in
the sense of soliton theory. Using these conditions, we prove non-integrability
for some scalar evolution PDEs of order . Also, we prove a result announced
in [arXiv:1303.3575] on the structure of the algebras for certain
classes of equations of orders , , , which include KdV, mKdV,
Kaup-Kupershmidt, Sawada-Kotera type equations. Among the obtained algebras for
equations considered in this paper and in [arXiv:1804.04652], one finds
infinite-dimensional Lie algebras of certain polynomial matrix-valued functions
on affine algebraic curves of genus and . In this approach, ZCRs may
depend on partial derivatives of arbitrary order, which may be higher than the
order of the equation . The algebras generalize Wahlquist-Estabrook
prolongation algebras, which are responsible for a much smaller class of ZCRs.Comment: 29 pages; v2: consideration of zero-curvature representations with
values in infinite-dimensional Lie algebras added. arXiv admin note: text
overlap with arXiv:1303.3575, arXiv:1804.04652, arXiv:1703.0721
On Lie algebras responsible for zero-curvature representations and Backlund transformations of (1+1)-dimensional scalar evolution PDEs
Zero-curvature representations (ZCRs) are one of the main tools in the theory
of integrable PDEs. In particular, Lax pairs for -dimensional PDEs can
be interpreted as ZCRs.
In [arXiv:1303.3575], for any -dimensional scalar evolution equation
, we defined a family of Lie algebras which are responsible for all
ZCRs of in the following sense. Representations of the algebras
classify all ZCRs of the equation up to local gauge transformations. Also,
using these algebras, one obtains necessary conditions for existence of a
Backlund transformation between two given equations. The algebras are
defined in [arXiv:1303.3575] in terms of generators and relations.
In this approach, ZCRs may depend on partial derivatives of arbitrary order,
which may be higher than the order of the equation . The algebras
generalize Wahlquist-Estabrook prolongation algebras, which are responsible for
a much smaller class of ZCRs.
In this preprint we prove a number of results on which were announced
in [arXiv:1303.3575]. We present applications of to the theory of
Backlund transformations in more detail and describe the explicit structure (up
to non-essential nilpotent ideals) of the algebras for a number of
equations of orders and .Comment: 40 pages. arXiv admin note: text overlap with arXiv:1303.357
Hydrodynamic-type systems describing 2-dimensional polynomially integrable geodesic flows
Starting from a homogeneous polynomial in momenta of arbitrary order we
extract multi-component hydrodynamic-type systems which describe 2-dimensional
geodesic flows admitting the initial polynomial as integral. All these
hydrodynamic-type systems are semi-Hamiltonian, thus implying that they are
integrable according to the generalized hodograph method. Moreover, they are
integrable in a constructive sense as polynomial first integrals allow to
construct generating equations of conservation laws. According to the
multiplicity of the roots of the polynomial integral, we separate integrable
particular cases
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