105 research outputs found
Global well-posedness for the Schroedinger equation coupled to a nonlinear oscillator
The Schroedinger equation with the nonlinearity concentrated at a single
point proves to be an interesting and important model for the analysis of
long-time behavior of solutions, such as the asymptotic stability of solitary
waves and properties of weak global attractors. In this note, we prove global
well-posedness of this system in the energy space H\sp 1.Comment: 11 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
Weighted Energy Decay for 3D Klein-Gordon Equation
We obtain a dispersive long-time decay in weighted energy norms for solutions
of the 3D Klein-Gordon equation with generic potential. The decay extends the
results obtained by Jensen and Kato for the 3D Schredinger equation. For the
proof we modify the spectral approach of Jensen and Kato to make it applicable
to relativistic equations
On Asymptotic Completeness of Scattering in the Nonlinear Lamb System, II
We establish the asymptotic completeness in the nonlinear Lamb system for
hyperbolic stationary states. For the proof we construct a trajectory of a
reduced equation (which is a nonlinear nonautonomous ODE) converging to a
hyperbolic stationary point using the Inverse Function Theorem in a Banach
space. We give the counterexamples showing nonexistence of such trajectories
for nonhyperbolic stationary points
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