1,617 research outputs found
A discrete linearizability test based on multiscale analysis
In this paper we consider the classification of dispersive linearizable partial difference equations defined on a quad-graph by the multiple scale reduction around their harmonic solution. We show that the A1, A2 and A3 linearizability conditions restrain the number of the parameters which enter into the equation. A subclass of the equations which pass the A3 C-integrability conditions can be linearized by a Möbius transformation
Simple and efficient LCAO basis sets for the diffuse states in carbon nanostructures
We present a simple way to describe the lowest unoccupied diffuse states in
carbon nanostructures in density functional theory (DFT) calculations using a
minimal LCAO (linear combination of atomic orbitals) basis set. By comparing
plane wave basis calculations, we show how these states can be captured by
adding long-range orbitals to the standard LCAO basis sets for the extreme
cases of planar sp2 (graphene) and curved carbon (C60). In particular,
using Bessel functions with a long range as additional basis functions retain a
minimal basis size. This provides a smaller and simpler atom-centered basis set
compared to the standard pseudo-atomic orbitals (PAOs) with multiple
polarization orbitals or by adding non-atom-centered states to the basis.Comment: 3 pages, 3 figure
Two novel classes of solvable many-body problems of goldfish type with constraints
Two novel classes of many-body models with nonlinear interactions "of
goldfish type" are introduced. They are solvable provided the initial data
satisfy a single constraint (in one case; in the other, two constraints): i.
e., for such initial data the solution of their initial-value problem can be
achieved via algebraic operations, such as finding the eigenvalues of given
matrices or equivalently the zeros of known polynomials. Entirely isochronous
versions of some of these models are also exhibited: i.e., versions of these
models whose nonsingular solutions are all completely periodic with the same
period.Comment: 30 pages, 2 figure
The Newtonian limit of the relativistic Boltzmann equation
The relativistic Boltzmann equation for a constant differential cross section
and with periodic boundary conditions is considered. The speed of light appears
as a parameter for a properly large and positive . A local
existence and uniqueness theorem is proved in an interval of time independent
of and conditions are given such that in the limit the
solutions converge, in a suitable norm, to the solutions of the
non-relativistic Boltzmann equation for hard spheres.Comment: 12 page
Lax pairs, Painlev\'e properties and exact solutions of the alogero Korteweg-de Vries equation and a new (2+1)-dimensional equation
We prove the existence of a Lax pair for the Calogero Korteweg-de Vries
(CKdV) equation. Moreover, we modify the T operator in the the Lax pair of the
CKdV equation, in the search of a (2+1)-dimensional case and thereby propose a
new equation in (2+1) dimensions. We named this the (2+1)-dimensional CKdV
equation. We show that the CKdV equation as well as the (2+1)-dimensional CKdV
equation are integrable in the sense that they possess the Painlev\'e property.
Some exact solutions are also constructed
Motion of Curves and Surfaces and Nonlinear Evolution Equations in (2+1) Dimensions
It is shown that a class of important integrable nonlinear evolution
equations in (2+1) dimensions can be associated with the motion of space curves
endowed with an extra spatial variable or equivalently, moving surfaces.
Geometrical invariants then define topological conserved quantities. Underlying
evolution equations are shown to be associated with a triad of linear
equations. Our examples include Ishimori equation and Myrzakulov equations
which are shown to be geometrically equivalent to Davey-Stewartson and Zakharov
-Strachan (2+1) dimensional nonlinear Schr\"odinger equations respectively.Comment: 13 pages, RevTeX, to appear in J. Math. Phy
On the simplest (2+1) dimensional integrable spin systems and their equivalent nonlinear Schr\"odinger equations
Using a moving space curve formalism, geometrical as well as gauge
equivalence between a (2+1) dimensional spin equation (M-I equation) and the
(2+1) dimensional nonlinear Schr\"odinger equation (NLSE) originally discovered
by Calogero, discussed then by Zakharov and recently rederived by Strachan,
have been estabilished. A compatible set of three linear equations are obtained
and integrals of motion are discussed. Through stereographic projection, the
M-I equation has been bilinearized and different types of solutions such as
line and curved solitons, breaking solitons, induced dromions, and domain wall
type solutions are presented. Breaking soliton solutions of (2+1) dimensional
NLSE have also been reported. Generalizations of the above spin equation are
discussed.Comment: 32 pages, no figures, accepted for publication in J. Math. Phy
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