18 research outputs found
On bosonic limits of two recent supersymmetric extensions of the Harry Dym hierarchy
Two generalized Harry Dym equations, recently found by Brunelli, Das and
Popowicz in the bosonic limit of new supersymmetric extensions of the Harry Dym
hierarchy [J. Math. Phys. 44:4756--4767 (2003)], are transformed into
previously known integrable systems: one--into a pair of decoupled KdV
equations, the other one--into a pair of coupled mKdV equations from a
bi-Hamiltonian hierarchy of Kupershmidt.Comment: 7 page
Classification of polynomial integrable systems of mixed scalar and vector evolution equations. I
We perform a classification of integrable systems of mixed scalar and vector
evolution equations with respect to higher symmetries. We consider polynomial
systems that are homogeneous under a suitable weighting of variables. This
paper deals with the KdV weighting, the Burgers (or potential KdV or modified
KdV) weighting, the Ibragimov-Shabat weighting and two unfamiliar weightings.
The case of other weightings will be studied in a subsequent paper. Making an
ansatz for undetermined coefficients and using a computer package for solving
bilinear algebraic systems, we give the complete lists of 2nd order systems
with a 3rd order or a 4th order symmetry and 3rd order systems with a 5th order
symmetry. For all but a few systems in the lists, we show that the system (or,
at least a subsystem of it) admits either a Lax representation or a linearizing
transformation. A thorough comparison with recent work of Foursov and Olver is
made.Comment: 60 pages, 6 tables; added one remark in section 4.2.17 (p.33) plus
several minor changes, to appear in J.Phys.
Local Isometric immersions of pseudo-spherical surfaces and evolution equations
The class of differential equations describing pseudo-spherical surfaces,
first introduced by Chern and Tenenblat [3], is characterized by the property
that to each solution of a differential equation, within the class, there
corresponds a 2-dimensional Riemannian metric of curvature equal to . The
class of differential equations describing pseudo-spherical surfaces carries
close ties to the property of complete integrability, as manifested by the
existence of infinite hierarchies of conservation laws and associated linear
problems. As such, it contains many important known examples of integrable
equations, like the sine-Gordon, Liouville and KdV equations. It also gives
rise to many new families of integrable equations. The question we address in
this paper concerns the local isometric immersion of pseudo-spherical surfaces
in from the perspective of the differential equations that give
rise to the metrics. Indeed, a classical theorem in the differential geometry
of surfaces states that any pseudo-spherical surface can be locally
isometrically immersed in . In the case of the sine-Gordon
equation, one can derive an expression for the second fundamental form of the
immersion that depends only on a jet of finite order of the solution of the
pde. A natural question is to know if this remarkable property extends to
equations other than the sine-Gordon equation within the class of differential
equations describing pseudo-spherical surfaces. In an earlier paper [11], we
have shown that this property fails to hold for all other second order
equations, except for those belonging to a very special class of evolution
equations. In the present paper, we consider a class of evolution equations for
of order describing pseudo-spherical surfaces. We show that
whenever an isometric immersion in exists, depending on a jet of
finite order of , then the coefficients of the second fundamental forms are
functions of the independent variables and only.Comment: Fields Institute Communications, 2015, Hamiltonian PDEs and
Applications, pp.N
A quasi-local mass for 2-spheres with negative Gauss curvature
We extend our previous definition of quasi-local mass to 2-spheres whose
Gauss curvature is negative and prove its positivity.Comment: 10 pages, Science in China, Series A: Math, to appea
On computer-assisted classification of coupled integrable equations
We show how the triangularization method of Moreno Maza can be successfully applied to the problem of classification of homogeneous coupled integrable equations. The classifications rely on the recent algorithm developed by Foursov that requires solving 17 systems of polynomial equations. We show that these systems can be completely resolved in the case of coupled Korteweg–de Vries, Sawada–Kotera and Kaup–Kupershmidt-type equations. c ○ 2002 Published by Elsevier Science Ltd
On computer-assisted classification of coupled integrable equations
We show how the triangularization method of the second author can be successfully applied to the problem of classification of homogeneous coupled integrable equations. The classifications rely on the recent algorithm developed by the first author that requires solving 17 systems of polynomial equations. We show that these systems can be completely resolved in the case of coupled Korteweg-de Vries, SawadaKotera and Kaup-Kupershmidt-type equations
Symmetry Structure of Integrable Nonevolutionary Equations
We study a class of evolutionary partial differential systems with two components related to second order (in time) nonevolutionary equations of odd order in spatial variable. We develop the formal diagonalization method in symbolic representation, which enables us to derive an explicit set of necessary conditions of existence of higher symmetries. Using these conditions we globally classify all such homogeneous integrable systems, i.e., systems which possess a hierarchy of infinitely many higher symmetries