27 research outputs found
Well-posedness, energy and charge conservation for nonlinear wave equations in discrete space-time
We consider the problem of discretization for the U(1)-invariant nonlinear
wave equations in any dimension. We show that the classical finite-difference
scheme used by Strauss and Vazquez \cite{MR0503140} conserves the
positive-definite discrete analog of the energy if the grid ratio is , where and are the mesh sizes of the time and space
variables and is the spatial dimension. We also show that if the grid ratio
is , then there is the discrete analog of the charge which is
conserved.
We prove the existence and uniqueness of solutions to the discrete Cauchy
problem. We use the energy conservation to obtain the a priori bounds for
finite energy solutions, thus showing that the Strauss -- Vazquez
finite-difference scheme for the nonlinear Klein-Gordon equation with positive
nonlinear term in the Hamiltonian is conditionally stable.Comment: 10 page
Polarons as stable solitary wave solutions to the Dirac-Coulomb system
We consider solitary wave solutions to the Dirac--Coulomb system both from
physical and mathematical points of view. Fermions interacting with gravity in
the Newtonian limit are described by the model of Dirac fermions with the
Coulomb attraction. This model also appears in certain condensed matter systems
with emergent Dirac fermions interacting via optical phonons. In this model,
the classical soliton solutions of equations of motion describe the physical
objects that may be called polarons, in analogy to the solutions of the
Choquard equation. We develop analytical methods for the Dirac--Coulomb system,
showing that the no-node gap solitons for sufficiently small values of charge
are linearly (spectrally) stable.Comment: Latex, 26 page
On spectral stability of solitary waves of nonlinear Dirac equation on a line
We study the spectral stability of solitary wave solutions to the nonlinear
Dirac equation in one dimension. We focus on the Dirac equation with cubic
nonlinearity, known as the Soler model in (1+1) dimensions and also as the
massive Gross-Neveu model. Presented numerical computations of the spectrum of
linearization at a solitary wave show that the solitary waves are spectrally
stable. We corroborate our results by finding explicit expressions for several
of the eigenfunctions. Some of the analytic results hold for the nonlinear
Dirac equation with generic nonlinearity.Comment: 20 pages with figure
Bifurcation and stability for Nonlinear Schroedinger equations with double well potential in the semiclassical limit
We consider the stationary solutions for a class of Schroedinger equations
with a symmetric double-well potential and a nonlinear perturbation. Here, in
the semiclassical limit we prove that the reduction to a finite-mode
approximation give the stationary solutions, up to an exponentially small term,
and that symmetry-breaking bifurcation occurs at a given value for the strength
of the nonlinear term. The kind of bifurcation picture only depends on the
non-linearity power. We then discuss the stability/instability properties of
each branch of the stationary solutions. Finally, we consider an explicit
one-dimensional toy model where the double well potential is given by means of
a couple of attractive Dirac's delta pointwise interactions.Comment: 46 pages, 4 figure
Stability and symmetry-breaking bifurcation for the ground states of a NLS with a interaction
We determine and study the ground states of a focusing Schr\"odinger equation
in dimension one with a power nonlinearity and a strong
inhomogeneity represented by a singular point perturbation, the so-called
(attractive) interaction, located at the origin. The
time-dependent problem turns out to be globally well posed in the subcritical
regime, and locally well posed in the supercritical and critical regime in the
appropriate energy space. The set of the (nonlinear) ground states is
completely determined. For any value of the nonlinearity power, it exhibits a
symmetry breaking bifurcation structure as a function of the frequency (i.e.,
the nonlinear eigenvalue) . More precisely, there exists a critical
value \om^* of the nonlinear eigenvalue \om, such that: if \om_0 < \om <
\om^*, then there is a single ground state and it is an odd function; if \om
> \om^* then there exist two non-symmetric ground states. We prove that before
bifurcation (i.e., for \om < \om^*) and for any subcritical power, every
ground state is orbitally stable. After bifurcation (\om =\om^*+0), ground
states are stable if does not exceed a value that lies
between 2 and 2.5, and become unstable for . Finally, for and \om \gg \om^*, all ground states are unstable. The branch of odd
ground states for \om \om^*,
obtaining a family of orbitally unstable stationary states. Existence of ground
states is proved by variational techniques, and the stability properties of
stationary states are investigated by means of the Grillakis-Shatah-Strauss
framework, where some non standard techniques have to be used to establish the
needed properties of linearization operators.Comment: 46 pages, 5 figure
Discrete peakons
We demonstrate for the first time the possibility for explicit construction
in a discrete Hamiltonian model of an exact solution of the form ,
i.e., a discrete peakon. These discrete analogs of the well-known, continuum
peakons of the Camassa-Holm equation [Phys. Rev. Lett. {\bf 71}, 1661 (1993)]
are found in a model different from their continuum siblings. Namely, we
observe discrete peakons in Klein-Gordon-type and nonlinear Schr\"odinger-type
chains with long-range interactions. The interesting linear stability
differences between these two chains are examined numerically and illustrated
analytically. Additionally, inter-site centered peakons are also obtained in
explicit form and their stability is studied. We also prove the global
well-posedness for the discrete Klein-Gordon equation, show the instability of
the peakon solution, and the possibility of a formation of a breathing peakon.Comment: Physica D, in pres
A system of ODEs for a Perturbation of a Minimal Mass Soliton
We study soliton solutions to a nonlinear Schrodinger equation with a
saturated nonlinearity. Such nonlinearities are known to possess minimal mass
soliton solutions. We consider a small perturbation of a minimal mass soliton,
and identify a system of ODEs similar to those from Comech and Pelinovsky
(2003), which model the behavior of the perturbation for short times. We then
provide numerical evidence that under this system of ODEs there are two
possible dynamical outcomes, which is in accord with the conclusions of
Pelinovsky, Afanasjev, and Kivshar (1996). For initial data which supports a
soliton structure, a generic initial perturbation oscillates around the stable
family of solitons. For initial data which is expected to disperse, the finite
dimensional dynamics follow the unstable portion of the soliton curve.Comment: Minor edit
Solitary waves in the Nonlinear Dirac Equation
In the present work, we consider the existence, stability, and dynamics of
solitary waves in the nonlinear Dirac equation. We start by introducing the
Soler model of self-interacting spinors, and discuss its localized waveforms in
one, two, and three spatial dimensions and the equations they satisfy. We
present the associated explicit solutions in one dimension and numerically
obtain their analogues in higher dimensions. The stability is subsequently
discussed from a theoretical perspective and then complemented with numerical
computations. Finally, the dynamics of the solutions is explored and compared
to its non-relativistic analogue, which is the nonlinear Schr{\"o}dinger
equation. A few special topics are also explored, including the discrete
variant of the nonlinear Dirac equation and its solitary wave properties, as
well as the PT-symmetric variant of the model
On asymptotic stability of ground states of some systems of nonlinear schr\uf6dinger equations
We extend to a specific class of systems of nonlinear Schr\uf6dinger equations (NLS) the theory of asymptotic stability of ground states already proved for the scalar NLS. Here the key point is the choice of an adequate system of modulation coordinates and the novelty, compared to the scalar NLS, is the fact that the group of symmetries of the system is non-commutative