1,512 research outputs found
On Darboux transformation of the supersymmetric sine-Gordon equation
Darboux transformation is constructed for superfields of the super
sine-Gordon equation and the superfields of the associated linear problem. The
Darboux transformation is shown to be related to the super B\"{a}cklund
transformation and is further used to obtain super soliton solutions.Comment: 9 Page
Non-polynomial extensions of solvable potentials a la Abraham-Moses
Abraham-Moses transformations, besides Darboux transformations, are
well-known procedures to generate extensions of solvable potentials in
one-dimensional quantum mechanics. Here we present the explicit forms of
infinitely many seed solutions for adding eigenstates at arbitrary real energy
through the Abraham-Moses transformations for typical solvable potentials, e.g.
the radial oscillator, the Darboux-P\"oschl-Teller and some others. These seed
solutions are simple generalisations of the virtual state wavefunctions, which
are obtained from the eigenfunctions by discrete symmetries of the potentials.
The virtual state wavefunctions have been an essential ingredient for
constructing multi-indexed Laguerre and Jacobi polynomials through multiple
Darboux-Crum transformations. In contrast to the Darboux transformations, the
virtual state wavefunctions generate non-polynomial extensions of solvable
potentials through the Abraham-Moses transformations.Comment: 29 page
On the algebraic invariant curves of plane polynomial differential systems
We consider a plane polynomial vector field of degree
. To each algebraic invariant curve of such a field we associate a compact
Riemann surface with the meromorphic differential . The
asymptotic estimate of the degree of an arbitrary algebraic invariant curve is
found. In the smooth case this estimate was already found by D. Cerveau and A.
Lins Neto [Ann. Inst. Fourier Grenoble 41, 883-903] in a different way.Comment: 10 pages, Latex, to appear in J.Phys.A:Math.Ge
Darboux theory of integrability for a class of nonautonomous vector fields
The goal of this paper is to extend the classical Darboux theory of integrability
from autonomous polynomial vector fields to a class of nonautonomous vector
fields. We also provide sufficient conditions for applying this theory of integrability
and we illustrate this theory in several examples.Postprint (published version
Darboux parameter for empty FRW quantum universes and quantum cosmological singularities
I present the factorization(s) of the Wheeler-DeWitt equation for vacuum FRW
minisuperspace universes of arbitrary Hartle-Hawking factor ordering, including
the so-called strictly isospectral supersymmetric method. By the latter means,
one can introduce an infinite class of singular FRW minisuperspace
wavefunctions characterized by a Darboux parameter that mathematically speaking
is a Riccati integration constant, while physically determines the position of
these strictly isospectral singularities on the Misner time axisComment: 3 pages, LaTe
A limitation of the hydrostatic reconstruction technique for Shallow Water equations
Because of their capability to preserve steady-states, well-balanced schemes
for Shallow Water equations are becoming popular. Among them, the hydrostatic
reconstruction proposed in Audusse et al. (2004), coupled with a positive
numerical flux, allows to verify important mathematical and physical properties
like the positivity of the water height and, thus, to avoid unstabilities when
dealing with dry zones. In this note, we prove that this method exhibits an
abnormal behavior for some combinations of slope, mesh size and water height.Comment: 7 page
Darboux Transformation of the Green Function for the Dirac Equation with the Generalized Potential
We consider the Darboux transformation of the Green functions of the regular
boundary problem of the one-dimensional stationary Dirac equation. We obtained
the Green functions of the transformed Dirac equation with the initial regular
boundary conditions. We also construct the formula for the unabridged trace of
the difference of the transformed and the initial Green functions of the
regular boundary problem of the one-dimensional stationary Dirac equation. We
illustrate our findings by the consideration of the Darboux transformation for
the Green function of the free particle Dirac equation on an interval.Comment: 14 pages,zip. file: Latex, 1 figure. Typos corrected, the figure
replace
Darboux Transformation for Dirac Equations with (1+1) potentials
We study the Darboux transformation (DT) for Dirac equations with (1+1)
potentials. Exact solutions for the adiabatic external field are constructed.
The connection between the exactly soluble Dirac (1+1) potentials and the
soliton solutions of the Davey--Stewartson equations is discussed.Comment: AMS-Te
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