74 research outputs found
The Zakharov-Shabat spectral problem on the semi-line: Hilbert formulation and applications
The inverse spectral transform for the Zakharov-Shabat equation on the
semi-line is reconsidered as a Hilbert problem. The boundary data induce an
essential singularity at large k to one of the basic solutions. Then solving
the inverse problem means solving a Hilbert problem with particular prescribed
behavior. It is demonstrated that the direct and inverse problems are solved in
a consistent way as soon as the spectral transform vanishes with 1/k at
infinity in the whole upper half plane (where it may possess single poles) and
is continuous and bounded on the real k-axis. The method is applied to
stimulated Raman scattering and sine-Gordon (light cone) for which it is
demonstrated that time evolution conserves the properties of the spectral
transform.Comment: LaTex file, 1 figure, submitted to J. Phys.
Darboux transformation for two component derivative nonlinear Schr\"odinger equation
In this paper, we consider the two component derivative nonlinear
Schr\"{o}dinger equation and present a simple Darboux transformation for it. By
iterating this Darboux transformation, we construct a compact representation
for the soliton solutions.Comment: 12 pages, 2 figure
Negaton and Positon solutions of the soliton equation with self-consistent sources
The KdV equation with self-consistent sources (KdVES) is used as a model to
illustrate the method. A generalized binary Darboux transformation (GBDT) with
an arbitrary time-dependent function for the KdVES as well as the formula for
-times repeated GBDT are presented. This GBDT provides non-auto-B\"{a}cklund
transformation between two KdV equations with different degrees of sources and
enable us to construct more general solutions with arbitrary -dependent
functions. By taking the special -function, we obtain multisoliton,
multipositon, multinegaton, multisoliton-positon, multinegaton-positon and
multisoliton-negaton solutions of KdVES. Some properties of these solutions are
discussed.Comment: 13 pages, Latex, no figues, to be published in J. Phys. A: Math. Ge
B\"{a}cklund transformations for the KP and mKP hierarchies with self-consistent sources
Using gauge transformations for the corresponding generating
pseudo-differential operators in terms of eigenfunctions and adjoint
eigenfunctions, we construct several types of auto-B\"{a}cklund transformations
for the KP hierarchy with self-consistent sources (KPHSCS) and mKP hierarchy
with self-consistent sources (mKPHSCS) respectively. The B\"{a}cklund
transformations from the KPHSCS to mKPHSCS are also constructed in this way.Comment: 22 pages. to appear in J.Phys.
Generalized Darboux transformations for the KP equation with self-consistent sources
The KP equation with self-consistent sources (KPESCS) is treated in the
framework of the constrained KP equation. This offers a natural way to obtain
the Lax representation for the KPESCS. Based on the conjugate Lax pairs, we
construct the generalized binary Darboux transformation with arbitrary
functions in time for the KPESCS which, in contrast with the binary Darboux
transformation of the KP equation, provides a non-auto-B\"{a}cklund
transformation between two KPESCSs with different degrees. The formula for
N-times repeated generalized binary Darboux transformation is proposed and
enables us to find the N-soliton solution and lump solution as well as some
other solutions of the KPESCS.Comment: 20 pages, no figure
Perturbation theory for nearly integrable multi-component nonlinear PDEs
The Riemann-Hilbert problem associated with the integrable PDE is used as a
nonlinear transformation of the nearly integrable PDE to the spectral space.
The temporal evolution of the spectral data is derived with account for
arbitrary perturbations and is given in the form of exact equations, which
generate the sequence of approximate ODEs in successive orders with respect to
the perturbation. For vector nearly integrable PDEs, embracing the vector NLS
and complex modified KdV equations, the main result is formulated in a theorem.
For a single vector soliton the evolution equations for the soliton parameters
and first-order radiation are given in explicit formComment: Submitted to Journal of Mathematical Physics (References are
corrected
The Solutions of the NLS Equations with Self-Consistent Sources
We construct the generalized Darboux transformation with arbitrary functions
in time for the AKNS equation with self-consistent sources (AKNSESCS)
which, in contrast with the Darboux transformation for the AKNS equation,
provides a non-auto-B\"{a}cklund transformation between two AKNSESCSs with
different degrees of sources. The formula for N-times repeated generalized
Darboux transformation is proposed. By reduction the generalized Darboux
transformation with arbitrary functions in time for the Nonlinear
Schr\"{o}dinger equation with self-consistent sources (NLSESCS) is obtained and
enables us to find the dark soliton, bright soliton and positon solutions for
NLSESCS and NLSESCS. The properties of these solution are analyzed.Comment: 24 pages, 3 figures, to appear in Journal of Physics A: Mathematical
and Genera
Vibrational Excitations in Weakly Coupled Single-Molecule Junctions: A Computational Analysis
In bulk systems, molecules are routinely identified by their vibrational
spectrum using Raman or infrared spectroscopy. In recent years, vibrational
excitation lines have been observed in low-temperature conductance measurements
on single molecule junctions and they can provide a similar means of
identification. We present a method to efficiently calculate these excitation
lines in weakly coupled, gateable single-molecule junctions, using a
combination of ab initio density functional theory and rate equations. Our
method takes transitions from excited to excited vibrational state into account
by evaluating the Franck-Condon factors for an arbitrary number of vibrational
quanta, and is therefore able to predict qualitatively different behaviour from
calculations limited to transitions from ground state to excited vibrational
state. We find that the vibrational spectrum is sensitive to the molecular
contact geometry and the charge state, and that it is generally necessary to
take more than one vibrational quantum into account. Quantitative comparison to
previously reported measurements on pi-conjugated molecules reveals that our
method is able to characterize the vibrational excitations and can be used to
identify single molecules in a junction. The method is computationally feasible
on commodity hardware.Comment: 9 pages, 7 figure
Algebraic construction of the Darboux matrix revisited
We present algebraic construction of Darboux matrices for 1+1-dimensional
integrable systems of nonlinear partial differential equations with a special
stress on the nonisospectral case. We discuss different approaches to the
Darboux-Backlund transformation, based on different lambda-dependencies of the
Darboux matrix: polynomial, sum of partial fractions, or the transfer matrix
form. We derive symmetric N-soliton formulas in the general case. The matrix
spectral parameter and dressing actions in loop groups are also discussed. We
describe reductions to twisted loop groups, unitary reductions, the matrix Lax
pair for the KdV equation and reductions of chiral models (harmonic maps) to
SU(n) and to Grassmann spaces. We show that in the KdV case the nilpotent
Darboux matrix generates the binary Darboux transformation. The paper is
intended as a review of known results (usually presented in a novel context)
but some new results are included as well, e.g., general compact formulas for
N-soliton surfaces and linear and bilinear constraints on the nonisospectral
Lax pair matrices which are preserved by Darboux transformations.Comment: Review paper (61 pages). To be published in the Special Issue
"Nonlinearity and Geometry: Connections with Integrability" of J. Phys. A:
Math. Theor. (2009), devoted to the subject of the Second Workshop on
Nonlinearity and Geometry ("Darboux Days"), Bedlewo, Poland (April 2008
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