News in Research: Prof. Dmitry Pelinovsky

• Fall-Winter, 2002: Applied Analysis: Evans function methods for algebraic solitons. During my visit and collaboration with Vassilis Rothos (Loughborough University, UK), we studied singularities of the Evans function in spectral problems associated with algebraically decaying solutions of nonlinear evolution equations. We proved that the singularities of the Evans function can be cancelled by a renormalization procedure. As a result, bifurcations of embedded eigenvalues and branch point bifurcations can be studied with Implicit Function Theorem applied to the renormalized Evans functions. Many delicate analytical results on this project were obtained later in 2003-2004 during my visit and collaboration with Martin Klaus (Virginia Tech, USA). General results are applied to the modified Korteweg- de Vries, focusing nonlinear Schrodinger, and massive Thiring equations. Applied to the modified KdV equation, our rigorous results verify formal perturbative results obtained in my earlier paper with R. Grimshaw. Download our paper published in Journal of Nonlinear Science.
• Spring-Fall, 2002: Asymptotic analysis: parametric resonances of dispersion-managed solitons. In our continuing collaboration with Jianke Yang (University of Vermont, USA), we developed asymptotic and numerical analysis of parametric resonance and radiative decay of dispersion-managed solitons in the dispersion-periodic NLS equation. Nonlinear Fermi golden rule is derived with the use of formal perturbation series expansions. The radiative decay of dispersion-managed solitons is exponential, algebraic and logarithmic at different stages of parametric resonance, in full agreement with direct numerical simulations. Download our paper published in SIAM Journal of Applied Mathematics.
• Spring-Fall, 2002: Spectral theory: gauge transformation and spectral decomposition for the Ishimori-II equations. During a visit of Derchyi Wu (Institute of Mathematics, Taiwan) to McMaster University, we have extended the inverse scattering transform for non-self-adjoint operators in two dimensions to the Ishimori-II equations. The Ishimori-II equations are related to Davey-Stewartson-II equations with the gauge transformation. Spectral decomposition theorem for Ishimori-II equations is proved under the commutativity between the gauge and adjoint transformations. Download our paper published in Journal of Physics A: Mathematical and General.
• Summer-Fall, 2002: Photorefractive optics: existence and stability of vortices and dipoles in the space of two dimensions. In our continuing collaborations with Jianke Yang (University of Vermont, USA), we systematically study existence and stability of vortex and dipole vector solitons in two coupled NLS equations with incoherent saturable nonlinearities. We develop formal perturbation expansions and show uniqueness of the family of vortex and dipole solitons near the bifurcation cut-off. All eigenvalues of linearized stability problem are traced near the bifurcation cut-off with the conclusion that both families of vortex and dipole solitons are stable near the bifurcation. We also predict instability of vortex solitons far from the bifurcation threshold by using theory of negative energy quadratic forms. Download our paper published in Physical Review E. Another paper was later developed in Fall-Winter, 2005 to analyze all symplectic rotations of coupled nonlinear Schrodinger equations with multi-vortex solutions. Download our paper published in Fundamentals and Applied Mathematics.