1,055 research outputs found
Global dynamics above the ground state for the nonlinear Klein-Gordon equation without a radial assumption
We extend our previous result on the focusing cubic Klein-Gordon equation in
three dimensions to the non-radial case, giving a complete classification of
global dynamics of all solutions with energy at most slightly above that of the
ground state.Comment: 40 page
Global attractor for a nonlinear oscillator coupled to the Klein-Gordon field
The long-time asymptotics is analyzed for all finite energy solutions to a
model U(1)-invariant nonlinear Klein-Gordon equation in one dimension, with the
nonlinearity concentrated at a single point: each finite energy solution
converges as time goes to plus or minus infinity to the set of all ``nonlinear
eigenfunctions'' of the form \psi(x)e\sp{-i\omega t}. The global attraction
is caused by the nonlinear energy transfer from lower harmonics to the
continuous spectrum and subsequent dispersive radiation.
We justify this mechanism by the following novel strategy based on inflation
of spectrum by the nonlinearity. We show that any omega-limit trajectory has
the time-spectrum in the spectral gap [-m,m] and satisfies the original
equation. This equation implies the key spectral inclusion for spectrum of the
nonlinear term. Then the application of the Titchmarsh Convolution Theorem
reduces the spectrum of each omega-limit trajectory to a single harmonic in
[-m,m].
The research is inspired by Bohr's postulate on quantum transitions and
Schroedinger's identification of the quantum stationary states to the nonlinear
eigenfunctions of the coupled U(1)-invariant Maxwell-Schroedinger and
Maxwell-Dirac equations.Comment: 29 pages, 1 figur
Noncommutative waves have infinite propagation speed
We prove the existence of global solutions to the Cauchy problem for
noncommutative nonlinear wave equations in arbitrary even spatial dimensions
where the noncommutativity is only in the spatial directions. We find that for
existence there are no conditions on the degree of the nonlinearity provided
the potential is positive. We furthermore prove that nonlinear noncommutative
waves have infinite propagation speed, i.e., if the initial conditions at time
0 have a compact support then for any positive time the support of the solution
can be arbitrarily large.Comment: 15 pages, references adde
Unique Continuation for Schr\"odinger Evolutions, with applications to profiles of concentration and traveling waves
We prove unique continuation properties for solutions of the evolution
Schr\"odinger equation with time dependent potentials. As an application of our
method we also obtain results concerning the possible concentration profiles of
blow up solutions and the possible profiles of the traveling waves solutions of
semi-linear Schr\"odinger equations.Comment: 23 page
A sharp condition for scattering of the radial 3d cubic nonlinear Schroedinger equation
We consider the problem of identifying sharp criteria under which radial
(finite energy) solutions to the focusing 3d cubic nonlinear
Schr\"odinger equation (NLS) scatter,
i.e. approach the solution to a linear Schr\"odinger equation as . The criteria is expressed in terms of the scale-invariant quantities
and , where denotes the
initial data, and and denote the (conserved in time) mass and
energy of the corresponding solution . The focusing NLS possesses a
soliton solution , where is the ground-state solution to a
nonlinear elliptic equation, and we prove that if and
, then the
solution is globally well-posed and scatters. This condition is sharp in
the sense that the soliton solution , for which equality in these
conditions is obtained, is global but does not scatter. We further show that if
, then the solution blows-up in finite time. The
technique employed is parallel to that employed by Kenig-Merle \cite{KM06a} in
their study of the energy-critical NLS
A sharp stability criterion for the Vlasov-Maxwell system
We consider the linear stability problem for a 3D cylindrically symmetric
equilibrium of the relativistic Vlasov-Maxwell system that describes a
collisionless plasma. For an equilibrium whose distribution function decreases
monotonically with the particle energy, we obtained a linear stability
criterion in our previous paper. Here we prove that this criterion is sharp;
that is, there would otherwise be an exponentially growing solution to the
linearized system. Therefore for the class of symmetric Vlasov-Maxwell
equilibria, we establish an energy principle for linear stability. We also
treat the considerably simpler periodic 1.5D case. The new formulation
introduced here is applicable as well to the nonrelativistic case, to other
symmetries, and to general equilibria
Compactness and existence results in weighted Sobolev spaces of radial functions. Part II: Existence
We prove existence and multiplicity results for finite energy solutions to
the nonlinear elliptic equation where is a radial domain (bounded or
unbounded) and satisfies on if and as
if is unbounded. The potential may be vanishing or unbounded at
zero or at infinity and the nonlinearity may be superlinear or sublinear.
If is sublinear, the case with is also considered.Comment: 29 pages, 8 figure
Anomalies of ac driven solitary waves with internal modes: Nonparametric resonances induced by parametric forces
We study the dynamics of kinks in the model subjected to a
parametric ac force, both with and without damping, as a paradigm of solitary
waves with internal modes. By using a collective coordinate approach, we find
that the parametric force has a non-parametric effect on the kink motion.
Specifically, we find that the internal mode leads to a resonance for
frequencies of the parametric driving close to its own frequency, in which case
the energy of the system grows as well as the width of the kink. These
predictions of the collective coordinate theory are verified by numerical
simulations of the full partial differential equation. We finally compare this
kind of resonance with that obtained for non-parametric ac forces and conclude
that the effect of ac drivings on solitary waves with internal modes is exactly
the opposite of their character in the partial differential equation.Comment: To appear in Phys Rev
On the Dynamics of solitons in the nonlinear Schroedinger equation
We study the behavior of the soliton solutions of the equation
i((\partial{\psi})/(\partialt))=-(1/(2m)){\Delta}{\psi}+(1/2)W_{{\epsilon}}'({\psi})+V(x){\psi}
where W_{{\epsilon}}' is a suitable nonlinear term which is singular for
{\epsilon}=0. We use the "strong" nonlinearity to obtain results on existence,
shape, stability and dynamics of the soliton. The main result of this paper
(Theorem 1) shows that for {\epsilon}\to0 the orbit of our soliton approaches
the orbit of a classical particle in a potential V(x).Comment: 29 page
Orbital stability: analysis meets geometry
We present an introduction to the orbital stability of relative equilibria of
Hamiltonian dynamical systems on (finite and infinite dimensional) Banach
spaces. A convenient formulation of the theory of Hamiltonian dynamics with
symmetry and the corresponding momentum maps is proposed that allows us to
highlight the interplay between (symplectic) geometry and (functional) analysis
in the proofs of orbital stability of relative equilibria via the so-called
energy-momentum method. The theory is illustrated with examples from finite
dimensional systems, as well as from Hamiltonian PDE's, such as solitons,
standing and plane waves for the nonlinear Schr{\"o}dinger equation, for the
wave equation, and for the Manakov system
- …