News in Research: Prof. Dmitry Pelinovsky

• Winter-Spring, 2005: Bose-Einstein condensates: Propagation of gap solitons in optical lattices under nonlinearity management. During my visit and collaboration with Mason Porter (Georgia Tech, USA), we studied a double-periodic problem, when gap solitons are supported by space-periodic potential (an optical lattice) and they are propagating in a nonlinear medium with sign-varying nonlinearity coefficient (Feshbach resonance management). We developed a double averaging technique to replace the full nonlinear Schrodinger (NLS) equation by the averaged NLS equation after time-averaging and to replace the averaged NLS equation by the coupled-mode system after space-averaging. Existence and stability of gap solitons in the resulting model, which is the coupled-mode system with symmetric fifth-order nonlinear functions, were studied with a number of analytical and numerical algorithms. Download our paper published in Physical Review E.
• Winter-Spring, 2005: Dynamical systems: Bifurcations of higher-order homoclinic orbits from higher-order zeros of Stokes constants. Initiated by visits and sabbatical work of Alex Tovbis (University of Central Florida, USA) in 2003-2004, we studied the fourth-order ordinary differential equation that arises for travelling wave solutions of the fifth-order Korteweg - de Vries equation with dispersive quadratic nonlinearities. Earlier numerical results suggested existence of a sequence of homoclinic orbits in the saddle-center system as a result of co-dimension one bifurcation. We proved this conjecture with an analytical theory, which relies on construction of outer and inner rigorous asymptotic solutions and analysis of Stokes constants at the truncated inner equation. Numerical algorithms of approximations of higher-order homoclinic orbits are developed on the grounds of rigorous theory. Download our paper published in Nonlinearity.
• Fall-Winter, 2004: Numerical analysis: Stability problem for gap solitons in coupled-mode equations. Together with my Ph.D. student Marina Chugunova (McMaster University, Canada), we found a block-diagonalization of the linearized stability problem for gap solitons within the system of coupled-mode equations with symmetric nonlinear functions. The block-diagonalization is used for effective numerical approximations of eigenvalues with smaller requirements on memory, faster computational speed, and higher numerical accuracy. Chebyshev interpolation algorithms and LAPACK eigenvalues solvers are applied to a number of practical computations for coupled-mode equations met in photonic crystals, nonlinear optics, and Bose-Einstein condensates with optical lattices. Download our paper published in SIAM Journal on Applied Dynamical Systems.
• Spring-Fall, 2004: Dynamical systems: Transformation of solitary waves of the Korteweg - de Vries equation due to nonlinear instability. During the visit of Andrew Comech (Texas A & M University, USA) and our further collaborations with him and Scipio Cuccagna (University of Reggio Emilia, Italy), we have studied nonlinear instability of critical solitary waves in the KdV equation with methods of central manifold reductions, spectral projections and analysis of remainder terms. Our work gives a rigorous ground to my earlier paper with R. Grimshaw, where we applied perturbation theory for solitons with radiative corrections. It turns out that this paper, which predicted correctly the rates for singularity formation in the critical KdV equation, can be rigorously verified with analysis of center manifold reductions and bounds on the nonlinear terms in exponentially weighted function spaces. Download our paper published in SIAM Journal of Mathematical Analysis.