184 research outputs found
On the analytical approach to the N-fold B\"acklund transformation of Davey-Stewartson equation
N-fold B\"acklund transformation for the Davey-Stewartson equation is
constructed by using the analytic structure of the Lax eigenfunction in the
complex eigenvalue plane. Explicit formulae can be obtained for a specified
value of N. Lastly it is shown how generalized soliton solutions are generated
from the trivial ones
Global well-posedness of the short-pulse and sine-Gordon equations in energy space
We prove global well-posedness of the short-pulse equation with small initial
data in Sobolev space . Our analysis relies on local well-posedness
results of Sch\"afer & Wayne, the correspondence of the short-pulse equation to
the sine-Gordon equation in characteristic coordinates, and a number of
conserved quantities of the short-pulse equation. We also prove local and
global well-posedness of the sine-Gordon equation in an appropriate function
space.Comment: 17 pages, revised versio
Generic solutions for some integrable lattice equations
We derive the expressions for -functions and generic solutions of
lattice principal chiral equations, lattice KP hierarchy and hierarchy
including lattice N-wave type equations. -function of free fermions
plays fundamental role in this context. Miwa's coordinates in our case appear
as the lattice parameters.Comment: The text of the talk at NEEDS-93 conference, Gallipoli, Italy,
September-93, LaTeX, 8 pages. Several typos and minor errors are correcte
Psi-series solutions of the cubic H\'{e}non-Heiles system and their convergence
The cubic H\'enon-Heiles system contains parameters, for most values of
which, the system is not integrable. In such parameter regimes, the general
solution is expressible in formal expansions about arbitrary movable branch
points, the so-called psi-series expansions. In this paper, the convergence of
known, as well as new, psi-series solutions on real time intervals is proved,
thereby establishing that the formal solutions are actual solutions
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
On integrability of a (2+1)-dimensional perturbed Kdv equation
A (2+1)-dimensional perturbed KdV equation, recently introduced by W.X. Ma
and B. Fuchssteiner, is proven to pass the Painlev\'e test for integrability
well, and its 44 Lax pair with two spectral parameters is found. The
results show that the Painlev\'e classification of coupled KdV equations by A.
Karasu should be revised
Darboux Transformations for a Lax Integrable System in -Dimensions
A -dimensional Lax integrable system is proposed by a set of specific
spectral problems. It contains Takasaki equations, the self-dual Yang-Mills
equations and its integrable hierarchy as examples. An explicit formulation of
Darboux transformations is established for this Lax integrable system. The
Vandermonde and generalized Cauchy determinant formulas lead to a description
for deriving explicit solutions and thus some rational and analytic solutions
are obtained.Comment: Latex, 14 pages, to be published in Lett. Math. Phy
From nonassociativity to solutions of the KP hierarchy
A recently observed relation between 'weakly nonassociative' algebras A (for
which the associator (A,A^2,A) vanishes) and the KP hierarchy (with dependent
variable in the middle nucleus A' of A) is recalled. For any such algebra there
is a nonassociative hierarchy of ODEs, the solutions of which determine
solutions of the KP hierarchy. In a special case, and with A' a matrix algebra,
this becomes a matrix Riccati hierarchy which is easily solved. The matrix
solution then leads to solutions of the scalar KP hierarchy. We discuss some
classes of solutions obtained in this way.Comment: 7 pages, 4 figures, International Colloquium 'Integrable Systems and
Quantum Symmetries', Prague, 15-17 June 200
Global well-posedness of the KP-I initial-value problem in the energy space
We prove that the KP-I initial value problem is globally well-posed in the
natural energy space of the equation
Breather lattice and its stabilization for the modified Korteweg-de Vries equation
We obtain an exact solution for the breather lattice solution of the modified
Korteweg-de Vries (MKdV) equation. Numerical simulation of the breather lattice
demonstrates its instability due to the breather-breather interaction. However,
such multi-breather structures can be stabilized through the concurrent
application of ac driving and viscous damping terms.Comment: 6 pages, 3 figures, Phys. Rev. E (in press
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