News in Research: Prof. Dmitry Pelinovsky

• Spring-Summer, 2004: Optical lattices: Discrete solitons and vortices in the discrete NLS equations. During my visit and collaborations with Panos Kevrekidis (University of Massachusetts, USA) and Dmitri Frantzeskakis (University of Athens, Greece), we studied persistence, multiplicity and stability of discrete solitons and vortices in the one-dimensional and two-dimensional NLS lattices. We proved the rigorous stability theorem for discrete solitons in the anti-continuous limit of the NLS lattice. The Lyapunov-Schmidt reductions are developed for the persistence and stability of discrete vortices in the same limit. We computed matrices for symmetric and asymmetric configurations of discrete vortices on the simplest closed discrete contours on the plane. Download our first paper published in Physica D. Download our second paper published in Physica D.
• Winter-Spring, 2004: Photonic crystals: Coupled-mode equations for low-contrast photonic crystals in three spatial dimensions. Together with my M.Sc. student Dmitri Agueev (McMaster University, Canada), we studied well-posedness of coupled-mode equations for multiple wave resonance in low-contrast, cubic-lattice, three-dimensional photonic crystals. New methods of analysis are exploited for linear stationary transmission problems that involve incident, transmitted, reflected and diffracted resonant Bloch waves. We study integer solutions of the resonance equation that defines the set of coupled-mode equations. We use the Floquet theory for Maxwell equations in three dimensions for asymptotic multi-scale reductions of the Maxwell equations to the coupled-mode equations. Convergence of generalized Fourier series solutions is proved with complex analysis and rigorous convergence theorems. Download our paper published in SIAM J. Appl. Math.
• Fall-Winter, 2003: Dynamical Systems: Normal forms for travelling breathers in the discrete NLS equations During my visit and collaboration with Vassilis Rothos (University of Leicester, UK), we studied the historical problem of existence of travelling breathers in the discrete NLS equation with cubic on-site nonlinearity. Unlike the previous numerical and heuristic papers, we developed the center manifold reductions and normal forms based on the formalism of G. Iooss. We derived the normal form at the maximum group velocity of the travelling breather, which is the NLS equation with the third-order dispersion. It follows from the third-order derivative NLS equation that no simple-humped travelling breather exists near the maximum group velocity, but a discrete infinite set of double-humped travelling breathers exists along families of one-parameter curves on the parameter plane, which accummulate to the bifurcation point. Download our paper published in Physica D.
• Spring-Fall, 2003: Dynamical systems: Diagonalization of spectral stability problem for KdV-solitons. During my visit and collaboration with Yuji Kodama (Ohio State University, USA), we studied N-solitons in the KdV hierachy, which includes the KdV and fifth-order KdV equations. We transformed the non-self-adjoint spectral problem with unbounded symplectic operator to the coupled problem for two self-adjoint operators in constrained function spaces. The number of unstable eigenvalues is found from the negative index of the quadratic forms, associated with the two operators, similar to the Morse theory for gradient dynamical systems. Download our paper published in J. Phys. A: Math. Gen.