445 research outputs found
Integration of a generalized H\'enon-Heiles Hamiltonian
The generalized H\'enon-Heiles Hamiltonian
with an additional
nonpolynomial term is known to be Liouville integrable for three
sets of values of . It has been previously integrated by genus
two theta functions only in one of these cases. Defining the separating
variables of the Hamilton-Jacobi equations, we succeed here, in the two other
cases, to integrate the equations of motion with hyperelliptic functions.Comment: LaTex 2e. To appear, Journal of Mathematical Physic
Theory of Pump Depletion and Spike Formation in Stimulated Raman Scattering
By using the inverse spectral transform, the SRS equations are solved and the
explicit output data is given for arbitrary laser pump and Stokes seed profiles
injected on a vacuum of optical phonons. For long duration laser pulses, this
solution is modified such as to take into account the damping rate of the
optical phonon wave. This model is used to interprete the experiments of Druhl,
Wenzel and Carlsten (Phys. Rev. Lett., (1983) vol. 51, p. 1171), in particular
the creation of a spike of (anomalous) pump radiation. The related nonlinear
Fourier spectrum does not contain discrete eigenvalue, hence this Raman spike
is not a soliton.Comment: LaTex file, includes two figures in LaTex format, 9 page
Optical Bistability in Nonlinear Optical Coupler with Negative Index Channel
We discuss a novel kind of nonlinear coupler with one channel filled with a
negative index material (NIM). The opposite directionality of the phase
velocity and the energy flow in the NIM channel facilitates an effective
feedback mechanism that leads to optical bistability and gap soliton formation
Controlled Generation of Dark Solitons with Phase Imprinting
The generation of dark solitons in Bose-Einstein condensates with phase
imprinting is studied by mapping it into the classic problem of a damped driven
pendulum. We provide simple but powerful schemes of designing the phase imprint
for various desired outcomes. We derive a formula for the number of dark
solitons generated by a given phase step, and also obtain results which explain
experimental observations.Comment: 4pages, 4 figure
INVERSE SCATTERING TRANSFORM ANALYSIS OF STOKES-ANTI-STOKES STIMULATED RAMAN SCATTERING
Zakharov-Shabat--Ablowitz-Kaup-Newel-Segur representation for
Stokes-anti-Stokes stimulated Raman scattering is proposed. Periodical waves,
solitons and self-similarity solutions are derived. Transient and bright
threshold solitons are discussed.Comment: 16 pages, LaTeX, no figure
Chiral Solitons in a Current Coupled Schr\"odinger Equation With Self Interaction
Recently non-topological chiral soliton solutions were obtained in a
derivatively coupled non-linear Schr\"odinger model in 1+1 dimensions. We
extend the analysis to include a more general self-coupling potential (which
includes the previous cases) and find chiral soliton solutions. Interestingly
even the magnitude of the velocity is found to be fixed. Energy and U(1) charge
associated with this non-topological chiral solitons are also obtained.Comment: 8 pages, no figure, to appear in Phys. Rev.
Q-stars and charged q-stars
We present the formalism of q-stars with local or global U(1) symmetry. The
equations we formulate are solved numerically and provide the main features of
the soliton star. We study its behavior when the symmetry is local in contrast
to the global case. A general result is that the soliton remains stable and
does not decay into free particles and the electrostatic repulsion preserves it
from gravitational collapse. We also investigate the case of a q-star with
non-minimal energy-momentum tensor and find that the soliton is stable even in
some cases of collapse when the coupling to gravity is absent.Comment: Latex, 19pg, 12 figures. Accepted in Phys. Rev.
Perturbative analysis of wave interactions in nonlinear systems
This work proposes a new way for handling obstacles to asymptotic
integrability in perturbed nonlinear PDEs within the method of Normal Forms -
NF - for the case of multi-wave solutions. Instead of including the whole
obstacle in the NF, only its resonant part is included, and the remainder is
assigned to the homological equation. This leaves the NF intergable and its
solutons retain the character of the solutions of the unperturbed equation. We
exploit the freedom in the expansion to construct canonical obstacles which are
confined to te interaction region of the waves. Fo soliton solutions, e.g., in
the KdV equation, the interaction region is a finite domain around the origin;
the canonical obstacles then do not generate secular terms in the homological
equation. When the interaction region is infifnite, or semi-infinite, e.g., in
wave-front solutions of the Burgers equation, the obstacles may contain
resonant terms. The obstacles generate waves of a new type, which cannot be
written as functionals of the solutions of the NF. When an obstacle contributes
a resonant term to the NF, this leads to a non-standard update of th wave
velocity.Comment: 13 pages, including 6 figure
Dissipative Boussinesq System of Equations in the B\'enard-Marangoni Phenomenon
By using the long-wave approximation, a system of coupled evolution equations
for the bulk velocity and the surface perturbations of a B\'enard-Marangoni
system is obtained. It includes nonlinearity, dispersion and dissipation, and
it can be interpreted as a dissipative generalization of the usual Boussinesq
system of equations. As a particular case, a strictly dissipative version of
the Boussinesq system is obtained. Finnaly, some speculations are made on the
nature of the physical phenomena described by this system of equations.Comment: 15 Pages, REVTEX (Version 3.0), no figure
On the (Non)-Integrability of KdV Hierarchy with Self-consistent Sources
Non-holonomic deformations of integrable equations of the KdV hierarchy are
studied by using the expansions over the so-called "squared solutions" (squared
eigenfunctions). Such deformations are equivalent to perturbed models with
external (self-consistent) sources. In this regard, the KdV6 equation is viewed
as a special perturbation of KdV equation. Applying expansions over the
symplectic basis of squared eigenfunctions, the integrability properties of the
KdV hierarchy with generic self-consistent sources are analyzed. This allows
one to formulate a set of conditions on the perturbation terms that preserve
the integrability. The perturbation corrections to the scattering data and to
the corresponding action-angle variables are studied. The analysis shows that
although many nontrivial solutions of KdV equations with generic
self-consistent sources can be obtained by the Inverse Scattering Transform
(IST), there are solutions that, in principle, can not be obtained via IST.
Examples are considered showing the complete integrability of KdV6 with
perturbations that preserve the eigenvalues time-independent. In another type
of examples the soliton solutions of the perturbed equations are presented
where the perturbed eigenvalue depends explicitly on time. Such equations,
however in general, are not completely integrable.Comment: 16 pages, no figures, LaTe
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