News in Research: Prof. Dmitry Pelinovsky
- Spring-Summer, 2003:
Spectral theory:
Spectra of positive and negative energies in the linearized NLS
problem. In a series of papers with Scipio
Cuccagna (University of Virginia, USA) and Vitali Vougalter (McMaster
University), we studied the spectrum of the linearized NLS problem
in three dimensions by rigorous spectral analysis. We proved a number
of statements on the relation between the number of negative eigenvalues
of the linearized Hamiltonian of the NLS equation and the number of
unstable eigenvalues of the spectral stability problem.
Bifurcations of resonances, embedded eigenvalues and multiple eigenvalues
from the stable spectrum are also classified in terms of the eigenvalues of
positive or negative energies. Download our paper published in Communications in Pure and Applied
Mathematics. Download another paper published in Journal of Mathematical Physics.
Download one more paper published in Journal of Mathematical Physics.
- Summer, 2003:
Bose-Einstein condensation:
Feshbach resonance and nonlinearity management of BEC solitons.
In the fast and dynamic collaboration with P. Kevrekidis (University
of Massachusetts, USA), we applied the averaging theory to the nonlinear
Schrodinger equation with periodically varying nonlinearity coefficient.
The averaging theory results in a local differential equation, which is
different from the integral equation for the dispersion-periodic
NLS equation. We studied existence of solitary wave solutions in
the local differential equation with numerical methods. Download
our paper published in Physical Review Letters. Since
an alternative method of Hamiltonian averaging leaded to
a different averaged equation,
we studied and compared all previously known averaging methods in application
to the nonlinearity management of BEC solitons in collaboration with V. Zharnitsky
(University of Illinois at Urbana-Champaign, USA). We discovered the failure of
the procedure reported in the PRL paper and corrected this error in the
brief report published in Physical Reviews. E. All details of
the different averaging methods are described in the review article published in the topical issue of
Chaos.
- Winter-Summer, 2003:
Spatial solitons:
Bifurcations of gap solitons in periodic potentials.
During my visit and collaboration with Yu. Kivshar (Australian
National University, Australia), we studied existence and
stability of gap solitons in the nonlinear Schrodinger equation
with the space-periodic potential. The physical problem describes the
Bose-Einstein condensates in periodic optical lattices.
We apply the asymptotic multi-scale expansion methods for
uniform solution of existence, multiplicity, bifurcation
and stability problems. Regular asymptotic series in powers of
small parameter are combined with exponential beyond-all-orders
asymptotic series. Download our paper published in Physical Reviews E.
- Spring-Summer, 2003: Spectral theory:
inertial laws for spectral stability of solitary waves.
During my visit and collaboration with Galina Perelman (Ecole Polytechnique,
France), I have rewritten my earlier preprint "Matrix stability
theory for incoherent optical solitons". The preprint was divided
into two papers, the first one gives rigorous analysis of Sylverster's
inertia theorems for spectral stability of solitary
waves and the second one features applications of analysis to
bifurcations and numerical computations for optical solitons.
Download the
first paper published in Proceedings of Royal Society of London A.
Download
the
second paper published in Studies in Applied Mathematics
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