4,646 research outputs found
Supersymmetric Modified Korteweg-de Vries Equation: Bilinear Approach
A proper bilinear form is proposed for the N=1 supersymmetric modified
Korteweg-de Vries equation. The bilinear B\"{a}cklund transformation of this
system is constructed. As applications, some solutions are presented for it.Comment: 8 pages, LaTeX using packages amsmath and amssymb, some corrections
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A survey of Hirota's difference equations
A review of selected topics in Hirota's bilinear difference equation (HBDE)
is given. This famous 3-dimensional difference equation is known to provide a
canonical integrable discretization for most important types of soliton
equations. Similarly to the continuous theory, HBDE is a member of an infinite
hierarchy. The central point of our exposition is a discrete version of the
zero curvature condition explicitly written in the form of discrete
Zakharov-Shabat equations for M-operators realized as difference or
pseudo-difference operators. A unified approach to various types of M-operators
and zero curvature representations is suggested. Different reductions of HBDE
to 2-dimensional equations are considered. Among them discrete counterparts of
the KdV, sine-Gordon, Toda chain, relativistic Toda chain and other typical
examples are discussed in detail.Comment: LaTeX, 43 pages, LaTeX figures (with emlines2.sty
Multiple addition theorem for discrete and continuous nonlinear problems
The addition relation for the Riemann theta functions and for its limits,
which lead to the appearance of exponential functions in soliton type equations
is discussed. The presented form of addition property resolves itself to the
factorization of N-tuple product of the shifted functions and it seems to be
useful for analysis of soliton type continuous and discrete processes in the
N+1 space-time. A close relation with the natural generalization of bi- and
tri-linear operators into multiple linear operators concludes the paper.Comment: 9 page
Complex Analysis of a Piece of Toda Lattice
We study a small piece of two dimensional Toda lattice as a complex dynamical
system. In particular the Julia set, which appears when the piece is deformed,
is shown analytically how it disappears as the system approaches to the
integrable limit.Comment: 17 pages, LaTe
An integrable generalization of the Toda law to the square lattice
We generalize the Toda lattice (or Toda chain) equation to the square
lattice; i.e., we construct an integrable nonlinear equation, for a scalar
field taking values on the square lattice and depending on a continuous (time)
variable, characterized by an exponential law of interaction in both discrete
directions of the square lattice. We construct the Darboux-Backlund
transformations for such lattice, and the corresponding formulas describing
their superposition. We finally use these Darboux-Backlund transformations to
generate examples of explicit solutions of exponential and rational type. The
exponential solutions describe the evolution of one and two smooth
two-dimensional shock waves on the square lattice.Comment: 14 pages, 4 figures, to appear in Phys. Rev. E http://pre.aps.org
Integrable dynamics of Toda-type on the square and triangular lattices
In a recent paper we constructed an integrable generalization of the Toda law
on the square lattice. In this paper we construct other examples of integrable
dynamics of Toda-type on the square lattice, as well as on the triangular
lattice, as nonlinear symmetries of the discrete Laplace equations on the
square and triangular lattices. We also construct the - function
formulations and the Darboux-B\"acklund transformations of these novel
dynamics.Comment: 22 pages, 4 figure
Temperature independent diffuse scattering and elastic lattice deformations in relaxor PbMg1/3Nb2/3O3
The results of diffuse neutron scattering experiment on PbMg1/3Nb2/3O3 single
crystal above the Burns temperature are reported. It is shown that the high
temperature elastic diffuse component is highly anisotropic in low-symmetry
Brillouin zones and this anisotropy can be described using Huang scattering
formalism assuming that the scattering originates from mesoscopic lattice
deformations due to elastic defects. The qualitative agreement between this
model and the experimental data is achieved with simple isotropic defects. It
is demonstrated that weak satellite maxima near the Bragg reflections can be
interpreted as the finite resolution effect.Comment: 7 pages, 7 figure
Invariant varieties of periodic points for some higher dimensional integrable maps
By studying various rational integrable maps on with
invariants, we show that periodic points form an invariant variety of dimension
for each period, in contrast to the case of nonintegrable maps in which
they are isolated. We prove the theorem: {\it `If there is an invariant variety
of periodic points of some period, there is no set of isolated periodic points
of other period in the map.'}Comment: 24 page
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