151 research outputs found
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
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
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
Anderson localization of one-dimensional hybrid particles
We solve the Anderson localization problem on a two-leg ladder by the
Fokker-Planck equation approach. The solution is exact in the weak disorder
limit at a fixed inter-chain coupling. The study is motivated by progress in
investigating the hybrid particles such as cavity polaritons. This application
corresponds to parametrically different intra-chain hopping integrals (a "fast"
chain coupled to a "slow" chain). We show that the canonical
Dorokhov-Mello-Pereyra-Kumar (DMPK) equation is insufficient for this problem.
Indeed, the angular variables describing the eigenvectors of the transmission
matrix enter into an extended DMPK equation in a non-trivial way, being
entangled with the two transmission eigenvalues. This extended DMPK equation is
solved analytically and the two Lyapunov exponents are obtained as functions of
the parameters of the disordered ladder. The main result of the paper is that
near the resonance energy, where the dispersion curves of the two decoupled and
disorder-free chains intersect, the localization properties of the ladder are
dominated by those of the slow chain. Away from the resonance they are
dominated by the fast chain: a local excitation on the slow chain may travel a
distance of the order of the localization length of the fast chain.Comment: 31 pages, 13 figure
Integrable dispersionless KdV hierarchy with sources
An integrable dispersionless KdV hierarchy with sources (dKdVHWS) is derived.
Lax pair equations and bi-Hamiltonian formulation for dKdVHWS are formulated.
Hodograph solution for the dispersionless KdV equation with sources (dKdVWS) is
obtained via hodograph transformation. Furthermore, the dispersionless
Gelfand-Dickey hierarchy with sources (dGDHWS) is presented.Comment: 15 pages, to be published in J. Phys. A: Math. Ge
Passive scalar convection in 2D long-range delta-correlated velocity field: Exact results
The letter presents new field-theoretical approach to 2D passive scalar
problem. The Gaussian form of the distribution for the Lyapunov exponent is
derived and its parameters are found explicitly.Comment: 11 pages, RevTex 3.0, IFUM-94/455/January-F
Bifurcation curves of subharmonic solutions
We revisit a problem considered by Chow and Hale on the existence of
subharmonic solutions for perturbed systems. In the analytic setting, under
more general (weaker) conditions, we prove their results on the existence of
bifurcation curves from the nonexistence to the existence of subharmonic
solutions. In particular our results apply also when one has degeneracy to
first order -- i.e. when the subharmonic Melnikov function vanishes
identically. Moreover we can deal as well with the case in which degeneracy
persists to arbitrarily high orders, in the sense that suitable generalisations
to higher orders of the subharmonic Melnikov function are also identically
zero. In general the bifurcation curves are not analytic, and even when they
are smooth they can form cusps at the origin: we say in this case that the
curves are degenerate as the corresponding tangent lines coincide. The
technique we use is completely different from that of Chow and Hale, and it is
essentially based on rigorous perturbation theory.Comment: 29 pages, 2 figure
On the Toda Lattice Equation with Self-Consistent Sources
The Toda lattice hierarchy with self-consistent sources and their Lax
representation are derived. We construct a forward Darboux transformation (FDT)
with arbitrary functions of time and a generalized forward Darboux
transformation (GFDT) for Toda lattice with self-consistent sources (TLSCS),
which can serve as a non-auto-Backlund transformation between TLSCS with
different degrees of sources. With the help of such DT, we can construct many
type of solutions to TLSCS, such as rational solution, solitons, positons,
negetons, and soliton-positons, soliton-negatons, positon-negatons etc., and
study properties and interactions of these solutions.Comment: 20 page
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.
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