News in Research: Prof. Dmitry Pelinovsky

• Summer-Fall, 2000: Partial differential equations: normal-form analysis of nonlinear resonances of embedded solitons. During my visit and collaboration with Jianke Yang (University of Vermont, USA), we proved fundamental results on nonlinear resonances of embedded solitons. The results include the normal form equation describing the semi-stability of perturbed embedded solitons in the second-harmonic-generating optical system. The proof is based on rigorous analysis of initial-value problems for PDEs and spectral decompositions for linear eigenvalue problems associated with soliton solutions. Download our paper published in Proc. Roy. Soc. Lond. A. Download another paper published in Wave Motion, where the semi-stability of embedded solitons is proved for the fifth-order KdV equation.
• Spring-Summer, 2000: Applied analysis: a final proof of stability-instability theorem for multi-parameter optical solitons. During my visit and collaboration with Yuri Kivshar (Australian National University, Australia), we developed the Vakhitov-Kolokolov' matrix method for analysis of stability of multi-parameter optical solitons. The one-to-one mapping of linear eigenvalue problems for differential operators to the matrix eigenvalue problems enabled us to prove the classical stability and instability theorems for multi-parameter solitons. The theorems generalized the remarkable 1990 stability theory by Grillakis, Shatah, and Strauss. Download our paper published in Phys. Rev. E. Download another paper published in Phys. Rev. Lett., where transverse instability of dipole-mode vector solitons is studied in space of two dimensions.
• Spring-Summer, 2000: Photonic optics: a first analytical proof of stability of optical limiters . The usually believed fact of stability of optical limiters is systematically proven. The proof is developed for the coupled-mode model of nonlinear optical gratings with the use of complex analysis. Download our paper published in J. Opt. Soc. Am. B and check the interactive research webpage for more information.
• Winter-Summer, 2000: Fiber optics communications: instability of 15-ps signals in soliton DCF modules! Rising at the global advances of network communication companies, my interest in dispersion-compensated optical fibers resulted in a new work. Propagation of soliton signals in the periodic nonlinear Schrodinger equation and emerging instabilities are discussed in a cool combination of analytical and numerical methods. Download my paper published in Phys. Rev. E and check the follow-up discussion on possible commercial applications.