845 research outputs found
Embedded Eigenvalues and the Nonlinear Schrodinger Equation
A common challenge to proving asymptotic stability of solitary waves is
understanding the spectrum of the operator associated with the linearized flow.
The existence of eigenvalues can inhibit the dispersive estimates key to
proving stability. Following the work of Marzuola & Simpson, we prove the
absence of embedded eigenvalues for a collection of nonlinear Schrodinger
equations, including some one and three dimensional supercritical equations,
and the three dimensional cubic-quintic equation. Our results also rule out
nonzero eigenvalues within the spectral gap and, in 3D, endpoint resonances.
The proof is computer assisted as it depends on the sign of certain inner
products which do not readily admit analytic representations. Our source code
is available for verification at
http://www.math.toronto.edu/simpson/files/spec_prop_asad_simpson_code.zip.Comment: 29 pages, 27 figures: fixed a typo in an equation from the previous
version, and added two equations to clarif
Stress dependent thermal pressurization of a fluid-saturated rock
Temperature increase in saturated porous materials under undrained conditions
leads to thermal pressurization of the pore fluid due to the discrepancy
between the thermal expansion coefficients of the pore fluid and of the solid
matrix. This increase in the pore fluid pressure induces a reduction of the
effective mean stress and can lead to shear failure or hydraulic fracturing.
The equations governing the phenomenon of thermal pressurization are presented
and this phenomenon is studied experimentally for a saturated granular rock in
an undrained heating test under constant isotropic stress. Careful analysis of
the effect of mechanical and thermal deformation of the drainage and pressure
measurement system is performed and a correction of the measured pore pressure
is introduced. The test results are modelled using a non-linear
thermo-poro-elastic constitutive model of the granular rock with emphasis on
the stress-dependent character of the rock compressibility. The effects of
stress and temperature on thermal pressurization observed in the tests are
correctly reproduced by the model
Fourier mode dynamics for the nonlinear Schroedinger equation in one-dimensional bounded domains
We analyze the 1D focusing nonlinear Schr\"{o}dinger equation in a finite
interval with homogeneous Dirichlet or Neumann boundary conditions. There are
two main dynamics, the collapse which is very fast and a slow cascade of
Fourier modes. For the cubic nonlinearity the calculations show no long term
energy exchange between Fourier modes as opposed to higher nonlinearities. This
slow dynamics is explained by fairly simple amplitude equations for the
resonant Fourier modes. Their solutions are well behaved so filtering high
frequencies prevents collapse. Finally these equations elucidate the unique
role of the zero mode for the Neumann boundary conditions
Stability of Discrete Solitons in the Presence of Parametric Driving
In this brief report, we consider parametrically driven bright solitons in
the vicinity of the anti-continuum limit. We illustrate the mechanism through
which these solitons become unstable due to the collision of the phase mode
with the continuous spectrum, or eigenvelues bifurcating thereof. We show how
this mechanism typically leads to complete destruction of the bright solitary
wave.Comment: 4 pages, 4 figure
A generalized nonlinear Schr\"odinger equation as model for turbulence, collapse, and inverse cascade
A two-dimensional generalized cubic nonlinear Schr\"odinger equation with
complex coefficients for the group dispersion and nonlinear terms is used to
investigate the evolution of a finite-amplitude localized initial perturbation.
It is found that modulation of the latter can lead to side-band formation, wave
condensation, collapse, turbulence, and inverse cascade, although not all
together nor in that order.Comment: 12 pages, 5 figure
Transport in simple networks described by integrable discrete nonlinear Schr\"Aodinger equation
We elucidate the case in which the Ablowitz-Ladik (AL) type discrete
nonlinear Schr\"Aodinger equa- tion (NLSE) on simple networks (e.g., star
graphs and tree graphs) becomes completely integrable just as in the case of a
simple 1-dimensional (1-d) discrete chain. The strength of cubic nonlinearity
is different from bond to bond, and networks are assumed to have at least two
semi-infinite bonds with one of them working as an incoming bond. The present
work is a nontrivial extension of our preceding one (Sobirov et al, Phys. Rev.
E 81, 066602 (2010)) on the continuum NLSE to the discrete case. We find: (1)
the solution on each bond is a part of the universal (bond-independent) AL
soliton solution on the 1-d discrete chain, but is multiplied by the inverse of
square root of bond-dependent nonlinearity; (2) nonlinearities at individual
bonds around each vertex must satisfy a sum rule; (3) under findings (1) and
(2), there exist an infinite number of constants of motion. As a practical
issue, with use of AL soliton injected through the incoming bond, we obtain
transmission probabilities inversely proportional to the strength of
nonlinearity on the outgoing bonds
Symmetry Breaking in Symmetric and Asymmetric Double-Well Potentials
Motivated by recent experimental studies of matter-waves and optical beams in
double well potentials, we study the solutions of the nonlinear Schr\"{o}dinger
equation in such a context. Using a Galerkin-type approach, we obtain a
detailed handle on the nonlinear solution branches of the problem, starting
from the corresponding linear ones and predict the relevant bifurcations of
solutions for both attractive and repulsive nonlinearities. The results
illustrate the nontrivial differences that arise between the steady
states/bifurcations emerging in symmetric and asymmetric double wells
Symbiotic Solitons in Heteronuclear Multicomponent Bose-Einstein condensates
We show that bright solitons exist in quasi-one dimensional heteronuclear
multicomponent Bose-Einstein condensates with repulsive self-interaction and
attractive inter-species interaction. They are remarkably robust to
perturbations of initial data and collisions and can be generated by the
mechanism of modulational instability. Some possibilities for control and the
behavior of the system in three dimensions are also discussed
Distribution of eigenfrequencies for oscillations of the ground state in the Thomas--Fermi limit
In this work, we present a systematic derivation of the distribution of
eigenfrequencies for oscillations of the ground state of a repulsive
Bose-Einstein condensate in the semi-classical (Thomas-Fermi) limit. Our
calculations are performed in 1-, 2- and 3-dimensional settings. Connections
with the earlier work of Stringari, with numerical computations, and with
theoretical expectations for invariant frequencies based on symmetry principles
are also given.Comment: 8 pages, 1 figur
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