323 research outputs found
The ILW hierarchy
In this paper, we present a new hierarchy which includes the intermediate
long wave (ILW) equation at the lowest order. This hierarchy is thought to be a
novel reduction of the 1st modified KP type hierarchy. The framework of our
investigation is Sato theory.Comment: 8 pages, LaTeX2
Explode-decay dromions in the non-isospectral Davey-Stewartson I (DSI) equation
In this letter, we report the existence of a novel type of explode-decay
dromions, which are exponentially localized coherent structures whose amplitude
varies with time, through Hirota method for a nonisospectral Davey-Stewartson
equation I discussed recently by Jiang. Using suitable transformations, we also
point out such solutions also exist for the isospectral Davey-Stewartson I
equation itself for a careful choice of the potentials
Backlund transformation and special solutions for Drinfeld-Sokolov-Satsuma-Hirota system of coupled equations
Using the Weiss method of truncated singular expansions, we construct an
explicit Backlund transformation of the Drinfeld-Sokolov-Satsuma-Hirota system
into itself. Then we find all the special solutions generated by this
transformation from the trivial zero solution of this system.Comment: LaTeX, 5 page
A new two-dimensional lattice model that is "consistent around a cube"
For two-dimensional lattice equations one definition of integrability is that
the model can be naturally and consistently extended to three dimensions, i.e.,
that it is "consistent around a cube" (CAC). As a consequence of CAC one can
construct a Lax pair for the model. Recently Adler, Bobenko and Suris conducted
a search based on this principle and certain additional assumptions. One of
those assumptions was the "tetrahedron property", which is satisfied by most
known equations. We present here one lattice equation that satisfies the
consistency condition but does not have the tetrahedron property. Its Lax pair
is also presented and some basic properties discussed.Comment: 8 pages in LaTe
Two-dimensional soliton cellular automaton of deautonomized Toda-type
A deautonomized version of the two-dimensional Toda lattice equation is
presented. Its ultra-discrete analogue and soliton solutions are also
discussed.Comment: 11 pages, LaTeX fil
Bilinear Discrete Painleve-II and its Particular Solutions
By analogy to the continuous Painlev\'e II equation, we present particular
solutions of the discrete Painlev\'e II (d-P) equation. These
solutions are of rational and special function (Airy) type. Our analysis is
based on the bilinear formalism that allows us to obtain the function
for d-P. Two different forms of bilinear d-P are obtained
and we show that they can be related by a simple gauge transformation.Comment: 9 pages in plain Te
A generalization of determinant formulas for the solutions of Painlev\'e II and XXXIV equations
A generalization of determinant formulas for the classical solutions of
Painlev\'e XXXIV and Painlev\'e II equations are constructed using the
technique of Darboux transformation and Hirota's bilinear formalism. It is
shown that the solutions admit determinant formulas even for the transcendental
case.Comment: 20 pages, LaTeX 2.09(IOP style), submitted to J. Phys.
Soliton formation from a pulse passing the zero-dispersion point in a nonlinear Schr\"odinger equation
We consider in detail the self-trapping of a soliton from a wave pulse that
passes from a defocussing region into a focussing one in a spatially
inhomogeneous nonlinear waveguide, described by a nonlinear Schrodinger
equation in which the dispersion coefficient changes its sign from normal to
anomalous. The model has direct applications to dispersion-decreasing nonlinear
optical fibers, and to natural waveguides for internal waves in the ocean. It
is found that, depending on the (conserved) energy and (nonconserved) mass of
the initial pulse, four qualitatively different outcomes of the pulse
transformation are possible: decay into radiation; self-trapping into a single
soliton; formation of a breather; and formation of a pair of counterpropagating
solitons. A corresponding chart is drawn on a parametric plane, which
demonstrates some unexpected features. In particular, it is found that any kind
of soliton(s) (including the breather and counterpropagating pair) eventually
decays into pure radiation with the increase of the energy, the initial mass
being kept constant. It is also noteworthy that a virtually direct transition
from a single soliton into a pair of symmetric counterpropagating ones seems
possible. An explanation for these features is proposed. In two cases when
analytical approximations apply, viz., a simple perturbation theory for broad
initial pulses, or the variational approximation for narrow ones, comparison
with the direct simulations shows reasonable agreement.Comment: 18 pages, 10 figures, 1 table. Phys. Rev. E, in pres
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