News in Research: Prof. Dmitry Pelinovsky

• Spring-Summer, 2005: Photorefractive crystals: Discrete vector vortices in the coupled discrete nonlinear Schrodinger lattices. In my continuing collaboration with Panos Kevrekidis (University of Massachusetts, USA), we applied our method of Lyapunov-Schmidt reductions to the on-site (cross-mode) vortices and their couplings. We found that the scalar discrete vortices and pairs of vector vortices are stable for small values of the coupling parameter below the Manakov limit. The pairs of vector solitons become however unstable for larger values of the coupling parameter. The paper gives an important practical information to physical groups, which try stimulating discrete vortices numerically and experimentally. Download our paper published in Proceeding of Royal Society of London A. The systematic symbolic computational package was developed in Summer, 2006 by my NSERC USRA student Mike Lukas. Download our paper submitted for publication.
• Spring-Summer, 2005: Water waves: Revised analysis of transverse instabilities of deep-water solitons. Initiated by the exchange at the SIAM conference on nonlinear waves in Orlando, USA (Ocrober, 2004). we have developed a collaboration with Bernard Deconinck (University of Washington, USA) and John Carter (Seattle University, USA). Based on their accurate numerical results, it became clear that the asymptotic predictions of my earlier paper are not accurate due to a miscalculation. Fixing the error, we developed analysis based on analytical properties of the Evans function continued accross the two branches of the continuous spectrum at the bifurcation point, where these two branches met. After a dynamical hunt for a correct asymptotic prediction, we have finally obtained a correct asymptotic law, verified both analytically and numerically. Download our paper published in Proceeding of Royal Society of London A.
• Winter-Spring, 2005: Dynamical Systems: Non-existence of travelling kinks in discrete phi-4 lattices. During my visit and collaboration with Igor Barashenkov (University of Cape Town, South Africa) and his student Oliver Oxtoby, we studied the historical problem of existence of standing and travelling kinks in various discretizations of the continuous phi-4 model. Previous expectional discretizations were developed by Speight and Ward (1994), Bender and Tovbis (1997), Flach, Zolotaryuk and Kladko (1999) and Kevrekidis (2003). These discretizations are exceptional since they provide existence of a continuous family of translationally invariant kinks, unlike the standard on-site nonlinear phi-4 lattice. We show however that all these discretizations fail to have generally continuous families of travelling kinks. Download our first paper published in Physical Review E. Download our second paper published in Nonlinearity.
• Winter-Spring, 2005: Bose-Einstein condensates: Stabilization of the time-periodic nonlinear Schrodinger equations in higher dimensions. During a short visit of Panos Kevrekidis (University of Massachusetts, USA), we started a number of projects, which evolved into new interesting publications. This project deals with the experimentally observed phenomenon of stabilization of multi-dimensional Bose-Einstein condensates under Feshbach resonance (nonlinearity management). With further involvement of Atanas Stefanov (University of Kansas, USA), we studied the averaged nonlinear Schrodinger equation, derived earlier for the nonlinearity management. We proved that blow-up of solutions of the averaged NLS equation, controlled by various norms of the conserved Hamiltonian, does not occur independently of the strength in the nonlinearity coefficient variance and the number of dimensions. Download our paper published in Journal of Physics A: Mathematical and General.