News in Research: Prof. Dmitry Pelinovsky

• Winter, 2004: Photonic crystal fibers: spatial vector solitons in two-dimensional fibers with defects. During my visit and collaboration with Jose Salgueiro and Yuri Kivshar (Australian National University, Australia), we studied existence and stability of two-dimensional vector solitons in a system of two coupled nonlinear Schrodinger equations with potentials that contain periodic and localized parts for photonic crystal fibers and defects. We show that the periodic structure can stabilize the spatial optical solitons, which would be unstable in a cubic two-dimensional system. Our brief paper features both numerical and asymptotic approaches to this applied problem. Download our paper published in the special issue of Studies in Applied Mathematics devoted to photonic crystals.
• Fall-Winter, 2004: Bose-Einstein Condensates: frequency and amplitude of oscillations of dark solitons in parabolic traps. During my visit and collaboration with Dimitri Franzeskakis (University of Athens, Greece), we addressed the recent contraversal problem of analytical theory of oscillations of dark solitons in Bose-Einstein condensates. We developed a version of perturbation theory for dark solitons and proved two main results: (i) the frequency of oscillations is independent of the soliton amplitude and (ii) the energy of dark solitons decays quadratically with velocity of dark solitons due to radiative losses. These results systematize the contradictory results obtained within various other alternative analytical techniques. Download our paper published in Physical Reviews E.
• Summer-Fall, 2004: Applied analysis: parameter resonance and decay of bound states under nonlinearity management. During the workshop on "Resonances in linear and nonlinear Schrodinger equations" (Fields Institute, Canada), we started a long-term project with E. Kirr (University of Chicago, USA) and S. Cuccagna (University of Reggio-Emilia, Italy) on rigorous analysis of the nonlinear Schrodinger equation with a periodically varying in time nonlinearity coefficient and localized in space potential. We gave a rigorous proof that the bound states (solitons) decay due to parametric resonance according to a nonlinear Fermi Golden rule. The method is based on consideration of small-norm bound states in three-dimensional NLS equation. The analysis uses spectral decompositions, contraction mapping principle and ODE comparison theory. Download our paper published in Journal of DIfferential Equations.
• Spring-Summer, 2004: Dynamical Systems: stability of embedded solitons in nonlinear optics. During my visit and collaboration with Jianke Yang (University of Vermont, USA), we studied a miraculous phenomenon of spectral stability of embedded solitons, which was previously discovered for a nonlinear Schrodinger equation with a third-order derivative term. We explained this phenomenon with the concept of resonant poles. A resonant pole bifurcates from a broken zero eigenvalue to the left half-plane due to the broken Hamiltonian structure of the NLS equation. Resonant poles correspond to exponentially growing functions, which can be detected with the techniques of exponential weights. Download our paper published in the special issue of Chaos on solitons in non-integrable systems.