News in Research: Prof. Dmitry Pelinovsky

• Summer-Fall, 2001: Applied analysis: optical bistability theorems for photonic gratings. During my research visit and collaborations with Arnd Scheel (Free University Berlin, Germany), we developed the Evans function methods for coupled-mode equations describing photonic gratings. The proof of optical bistability theory postulated in the end of 70s is finally developed on the level of Taylor expansions of the Evans function. We have discovered that instabilities of stationary solutions in finite and infinite-interval structures resemble instabilities of dissipative dynamical systems. Computations of the winding number for the Evans function are performed to study mechanisms of bistability and Hopf bifurcations. Download our paper published in J. Nonlin. Sci.
• Spring-Fall, 2001: Spectral theory : revisited proofs of matrix stability-instability theorems. Reductions of a non-self-adjoint linearized stability problem for optical solitons to a pair of uncoupled constrained problems for self-adjoint matrix Schrodinger operators are rigorously studied. The proof of matrix stability-instability theorems is based on linear algebra theorems on simalteneous diagonalization of two linear self-adjoint operators. Complex instabilities and transverse instabilities are utilized within the uniform approach. Sharp bounds on number and type of unstable eigenvalues of the linearized problem are derived in the new method. The new computational algorithms is proposed as the alternative to the winding number computations of the Evans (analytical) function. The new method reduces numerical complexity of the linear stability problem for optical solitons. Download my preprint which was never published.
• Summer-Fall, 2001: Fiber optics communications: averaging methods for dispersion-managed pulses. The first-order averaging theory is developed in collaborations with Vadim Zharnitsky (Bell Laboratories, USA). All bifurcation curves for existence and stability of dispersion-managed pulses are computed analytically for a low-dimensional system in the Gaussian approximation. Convergence of the first-order averaging theory is proved and errors of approximations are controlled for an infinite-dimensional reduction of a periodic NLS equation to an averaged integral NLS equation. The results finally explain the mystery of two branches of dispersion-managed pulses in the normal-dispersion regime of fiber optics networks! Download our paper published in SIAM J. Appl. Math. Download another paper with Henrik Kalisch published in Steklov Mathematical Institute Proceedings Supplement, where asymptotics of dispersion-managed solitons are studied in the limit of large pulse energy and dispersion map strength.
• Winter-Spring, 2001: Photonic optics: an unconditionally stable numerical algorithm for optical limiters. In collaborations with Ted Sargent's group at ECE department of University of Toronto, we have developed the MATLAB software package for simulating nonlinear periodic optical gratings. Stability of optical limiters, proven first analytically, is confirmed by accurate numerical computations. The software package is being used for extensive numerical computations in the group's application-based research. Download our paper published in J. Opt. Soc. Am. B. Download another paper, published in J. Opt. Soc. Am. B., where transmission of optical pulses is studied for optical limiters.