63 research outputs found
B\"acklund transformations for the second Painlev\'e hierarchy: a modified truncation approach
The second Painlev\'e hierarchy is defined as the hierarchy of ordinary
differential equations obtained by similarity reduction from the modified
Korteweg-de Vries hierarchy. Its first member is the well-known second
Painlev\'e equation, P2.
In this paper we use this hierarchy in order to illustrate our application of
the truncation procedure in Painlev\'e analysis to ordinary differential
equations. We extend these techniques in order to derive auto-B\"acklund
transformations for the second Painlev\'e hierarchy. We also derive a number of
other B\"acklund transformations, including a B\"acklund transformation onto a
hierarchy of P34 equations, and a little known B\"acklund transformation for P2
itself.
We then use our results on B\"acklund transformations to obtain, for each
member of the P2 hierarchy, a sequence of special integrals.Comment: 12 pages in LaTeX 2.09 (uses ioplppt.sty), to appear in Inverse
Problem
Non-classical symmetries and the singular manifold method: A further two examples
This paper discusses two equations with the conditional Painleve property.
The usefulness of the singular manifold method as a tool for determining the
non-classical symmetries that reduce the equations to ordinary differential
equations with the Painleve property is confirmed once moreComment: 9 pages (latex), to appear in Journal of Physics
Mappings preserving locations of movable poles: a new extension of the truncation method to ordinary differential equations
The truncation method is a collective name for techniques that arise from
truncating a Laurent series expansion (with leading term) of generic solutions
of nonlinear partial differential equations (PDEs). Despite its utility in
finding Backlund transformations and other remarkable properties of integrable
PDEs, it has not been generally extended to ordinary differential equations
(ODEs). Here we give a new general method that provides such an extension and
show how to apply it to the classical nonlinear ODEs called the Painleve
equations. Our main new idea is to consider mappings that preserve the
locations of a natural subset of the movable poles admitted by the equation. In
this way we are able to recover all known fundamental Backlund transformations
for the equations considered. We are also able to derive Backlund
transformations onto other ODEs in the Painleve classification.Comment: To appear in Nonlinearity (22 pages
Existence and Uniqueness of Tri-tronqu\'ee Solutions of the second Painlev\'e hierarchy
The first five classical Painlev\'e equations are known to have solutions
described by divergent asymptotic power series near infinity. Here we prove
that such solutions also exist for the infinite hierarchy of equations
associated with the second Painlev\'e equation. Moreover we prove that these
are unique in certain sectors near infinity.Comment: 13 pages, Late
Symmetries of a class of nonlinear fourth order partial differential equations
In this paper we study symmetry reductions of a class of nonlinear fourth
order partial differential equations \be u_{tt} = \left(\kappa u + \gamma
u^2\right)_{xx} + u u_{xxxx} +\mu u_{xxtt}+\alpha u_x u_{xxx} + \beta u_{xx}^2,
\ee where , , , and are constants. This
equation may be thought of as a fourth order analogue of a generalization of
the Camassa-Holm equation, about which there has been considerable recent
interest. Further equation (1) is a ``Boussinesq-type'' equation which arises
as a model of vibrations of an anharmonic mass-spring chain and admits both
``compacton'' and conventional solitons. A catalogue of symmetry reductions for
equation (1) is obtained using the classical Lie method and the nonclassical
method due to Bluman and Cole. In particular we obtain several reductions using
the nonclassical method which are no} obtainable through the classical method
The Hamiltonian Structure of the Second Painleve Hierarchy
In this paper we study the Hamiltonian structure of the second Painleve
hierarchy, an infinite sequence of nonlinear ordinary differential equations
containing PII as its simplest equation. The n-th element of the hierarchy is a
non linear ODE of order 2n in the independent variable depending on n
parameters denoted by and . We introduce new
canonical coordinates and obtain Hamiltonians for the and
evolutions. We give explicit formulae for these Hamiltonians showing that they
are polynomials in our canonical coordinates
Analytic and Asymptotic Methods for Nonlinear Singularity Analysis: a Review and Extensions of Tests for the Painlev\'e Property
The integrability (solvability via an associated single-valued linear
problem) of a differential equation is closely related to the singularity
structure of its solutions. In particular, there is strong evidence that all
integrable equations have the Painlev\'e property, that is, all solutions are
single-valued around all movable singularities. In this expository article, we
review methods for analysing such singularity structure. In particular, we
describe well known techniques of nonlinear regular-singular-type analysis,
i.e. the Painlev\'e tests for ordinary and partial differential equations. Then
we discuss methods of obtaining sufficiency conditions for the Painlev\'e
property. Recently, extensions of \textit{irregular} singularity analysis to
nonlinear equations have been achieved. Also, new asymptotic limits of
differential equations preserving the Painlev\'e property have been found. We
discuss these also.Comment: 40 pages in LaTeX2e. To appear in the Proceedings of the CIMPA Summer
School on "Nonlinear Systems," Pondicherry, India, January 1996, (eds) B.
Grammaticos and K. Tamizhman
Painlev\'{e} test of coupled Gross-Pitaevskii equations
Painlev\'{e} test of the coupled Gross-Pitaevskii equations has been carried
out with the result that the coupled equations pass the P-test only if a
special relation containing system parameters (masses, scattering lengths) is
satisfied. Computer algebra is applied to evaluate j=4 compatibility condition
for admissible external potentials. Appearance of an arbitrary real potential
embedded in the external potentials is shown to be the consequence of the
coupling. Connection with recent experiments related to stability of
two-component Bose-Einstein condensates of Rb atoms is discussed.Comment: 13 pages, no figure
Lattice modified KdV hierarchy from a Lax pair expansion
We produce a hierarchiy of integrable equations by systematically adding
terms to the Lax pair for the lattice modified KdV equation. The equations in
the hierarchy are related to one aonother by recursion relations. These
recursion relations are solved explicitly so that every equation in the
hierarchy along with its Lax pair is known
Integrable discretizations of derivative nonlinear Schroedinger equations
We propose integrable discretizations of derivative nonlinear Schroedinger
(DNLS) equations such as the Kaup-Newell equation, the Chen-Lee-Liu equation
and the Gerdjikov-Ivanov equation by constructing Lax pairs. The discrete DNLS
systems admit the reduction of complex conjugation between two dependent
variables and possess bi-Hamiltonian structure. Through transformations of
variables and reductions, we obtain novel integrable discretizations of the
nonlinear Schroedinger (NLS), modified KdV (mKdV), mixed NLS, matrix NLS,
matrix KdV, matrix mKdV, coupled NLS, coupled Hirota, coupled Sasa-Satsuma and
Burgers equations. We also discuss integrable discretizations of the
sine-Gordon equation, the massive Thirring model and their generalizations.Comment: 24 pages, LaTeX2e (IOP style), final versio
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