351 research outputs found
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
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
Global Attraction to Solitary Waves in Models Based on the Klein-Gordon Equation
We review recent results on global attractors of U(1)-invariant dispersive
Hamiltonian systems. We study several models based on the Klein-Gordon equation
and sketch the proof that in these models, under certain generic assumptions,
the weak global attractor is represented by the set of all solitary waves. In
general, the attractors may also contain multifrequency solitary waves; we give
examples of systems which contain such solutions.Comment: This is a contribution to the Proc. of the Seventh International
Conference ''Symmetry in Nonlinear Mathematical Physics'' (June 24-30, 2007,
Kyiv, Ukraine), published in SIGMA (Symmetry, Integrability and Geometry:
Methods and Applications) at http://www.emis.de/journals/SIGMA
Global Attraction to Solitary Waves in Models Based on the Klein-Gordon Equation
We review recent results on global attractors of U(1)-invariant dispersive
Hamiltonian systems. We study several models based on the Klein-Gordon equation
and sketch the proof that in these models, under certain generic assumptions,
the weak global attractor is represented by the set of all solitary waves. In
general, the attractors may also contain multifrequency solitary waves; we give
examples of systems which contain such solutions.Comment: This is a contribution to the Proc. of the Seventh International
Conference ''Symmetry in Nonlinear Mathematical Physics'' (June 24-30, 2007,
Kyiv, Ukraine), published in SIGMA (Symmetry, Integrability and Geometry:
Methods and Applications) at http://www.emis.de/journals/SIGMA
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