9,089 research outputs found
The Effect of low Momentum Quantum Fluctuations on a Coherent Field Structure
In the present work the evolution of a coherent field structure of the
Sine-Gordon equation under quantum fluctuations is studied. The basic equations
are derived from the coherent state approximation to the functional
Schr\"odinger equation for the field. These equations are solved asymptotically
and numerically for three physical situations. The first is the study of the
nonlinear mechanism responsible for the quantum stability of the soliton in the
presence of low momentum fluctuations. The second considers the scattering of a
wave by the Soliton. Finally the third problem considered is the collision of
Solitons and the stability of a breather.
It is shown that the complete integrability of the Sine-Gordon equation
precludes fusion and splitting processes in this simplified model.
The approximate results obtained are non-perturbative in nature, and are
valid for the full nonlinear interaction in the limit of low momentum
fluctuations. It is also found that these approximate results are in good
agreement with full numerical solutions of the governing equations. This
suggests that a similar approach could be used for the baby Skyrme model, which
is not completely integrable. In this case the higher space dimensionality and
the internal degrees of freedom which prevent the integrability will be
responsable for fusion and splitting processes. This work provides a starting
point in the numerical solution of the full quantum problem of the interaction
of the field with a fluctuation.Comment: 15 pages, 9 (ps) figures, Revtex file. Some discussion expanded but
conclusions unchanged. Final version to appear in PR
On the evolution of scattering data under perturbations of the Toda lattice
We present the results of an analytical and numerical study of the long-time
behavior for certain Fermi-Pasta-Ulam (FPU) lattices viewed as perturbations of
the completely integrable Toda lattice. Our main tools are the direct and
inverse scattering transforms for doubly-infinite Jacobi matrices, which are
well-known to linearize the Toda flow. We focus in particular on the evolution
of the associated scattering data under the perturbed vs. the unperturbed
equations. We find that the eigenvalues present initially in the scattering
data converge to new, slightly perturbed eigenvalues under the perturbed
dynamics of the lattice equation. To these eigenvalues correspond solitary
waves that emerge from the solitons in the initial data. We also find that new
eigenvalues emerge from the continuous spectrum as the lattice system is let to
evolve under the perturbed dynamics.Comment: 27 pages, 17 figures. Revised Introduction and Discussion section
Numerical Solitons of Generalized Korteweg-de Vries Equations
We propose a numerical method for finding solitary wave solutions of
generalized Korteweg-de Vries equations by solving the nonlinear eigenvalue
problem on an unbounded domain. The artificial boundary conditions are obtained
to make the domain finite. We specially discuss the soliton solutions of the
K(m, n) equation and KdV-K(m,n) equation. Furthermore for the mixed models of
linear and nonlinear dispersion, the collision behaviors of soliton-soliton and
soliton-antisoliton are observed.Comment: 9 pages, 4 figure
Macroscopic dynamics of incoherent soliton ensembles: soliton-gas kinetics and direct numerical modeling
We undertake a detailed comparison of the results of direct numerical
simulations of the integrable soliton gas dynamics with the analytical
predictions inferred from the exact solutions of the relevant kinetic equation
for solitons. We use the KdV soliton gas as a simplest analytically accessible
model yielding major insight into the general properties of soliton gases in
integrable systems. Two model problems are considered: (i) the propagation of a
`trial' soliton through a one-component `cold' soliton gas consisting of
randomly distributed solitons of approximately the same amplitude; and (ii)
collision of two cold soliton gases of different amplitudes (soliton gas shock
tube problem) leading to the formation of an incoherend dispersive shock wave.
In both cases excellent agreement is observed between the analytical
predictions of the soliton gas kinetics and the direct numerical simulations.
Our results confirm relevance of the kinetic equation for solitons as a
quantitatively accurate model for macroscopic non-equilibrium dynamics of
incoherent soliton ensembles.Comment: 20 pages, 8 figures, 34 references. Other author's papers can be
downloaded at http://www.denys-dutykh.com
Stabilization of solitons of the multidimensional nonlinear Schrodinger equation: Matter-wave breathers
We demonstrate that stabilization of solitons of the multidimensional
Schrodinger equation with a cubic nonlinearity may be achieved by a suitable
periodic control of the nonlinear term. The effect of this control is to
stabilize the unstable solitary waves which belong to the frontier between
expanding and collapsing solutions and to provide an oscillating solitonic
structure, some sort of breather-type solution. We obtain precise conditions on
the control parameters to achieve the stabilization and compare our results
with accurate numerical simulations of the nonlinear Schrodinger equation.
Because of the application of these ideas to matter waves these solutions are
some sort of matter breathers
- …