News in Research: Prof. Dmitry Pelinovsky

• Winter-Spring, 2006: Difference Equations: Classification of translationally invariant discrete nonlinear Schrodinger equations. I have considered the ten-parameter extension of the discrete NLS equation, which represents the most general reversible cubic NLS lattice. It is shown that the six-parameter nonlinearity preserves a reduction of the second-order difference equation for stationary solutions to the first-order difference equation. Furthermore, the four-parameter nonlinearity preserves an additional constraint that all stationary solutions are real-valued for small lattice spacing. Finally, the three-parameter nonlinearity survives the reduction of the differential advance-delay equation for traveling solutions to the third-order normal form which has localized single-humped solutions. The final discretization is a two-parameter extension of the Ablowitz-Ladik lattice, which may admit non-trivial discrete traveling solutions. Download my paper published in Nonlinearity.
• Fall-Winter, 2005: Dynamical systems: Numerical approximations and spectral stability of two-pulse solutions in the fifth-order Korteweg-de Vries equation. In collaboration with my summer student M. Fioroni and my Ph.D. student M. Chugunova, we have considered two problems related to two-pulse solutions of the fifth-order KdV model. When a root finding algorithm is added to divergent iterations of the Petviashvili method, the two-pulse solutions can be approximated numerically with the spectral accuracy. The stability of two-pulse solutions is studied with the method of Lyapunov--Schmidt reductions, the count of eigenvalues in Pontryagin spaces and analysis of exponential weighted spaces. In particular, we prove persistence of embedded eigenvalues related to the spectrally stable two-pulse solutions. Download our paper submitted to Discrete and Continuous Dynamical Systems Series B.
• Summer-Fall, 2005: Dynamical Systems: Normal forms for travelling kinks in discrete Klein-Gordon lattices. During my visit and collaboration with Gerard Iooss (Institute de Nonlinearity, France), we analyzed heteroclinic orbits of discrete Klein-Gordon lattices with rigorous theory of center manifold reductions and normal forms. We derived the normal form equation (the fourth-order ODE) for a particular point on the parameter plane of travelling kink, which corresponds to a quadruple zero eigenvalue of a linearization operator. Non-existence of heteroclinic orbits for single kinks and existence of such orbits for multiple kinks are proved within the normal form by a systematic numerical procedure. Download our paper published in Physica D.
• Summer-Fall, 2005: Spectral theory: Count of eigenvalues in Pontryagin space. In collaboration with my Ph.D. student Marina Chugunova (McMaster University, Canada), we have revised the theory of L.S. Pontryagin, M.G. Krein and M. Grillakis for spectral stability problems related to nonlinear evolution equations. This revision is two-fold. First we have recovered the recently obtained theorems on negative indices of linearized Hamiltonians and unstable eigenvalues of stability problems with a simple and elegant computation. Second we have found new bounds on the number of imaginary eigenvalues of positive Krein singatures in terms of positive eigenvalues of the linearized Hamiltonian. Download our paper submitted to SIAM Journal of Mathematical Analysis.