News in Research: Prof. Dmitry Pelinovsky

• Summer, 2002: Fluid dynamics: generation of multiple large-amplitude interval waves. During my visits to Loughborough University (UK) and Institute of Applied Physics (Russia), I have attended a historical problem of generation of solitons in the Gardner equation with negative cubic nonlinear term and a positive dispersion term. In this joint collaboration with my Ph.D. advisor Roger Grimshaw, my father Efim Pelinovsky, and his former student Alexey Slunyaev, we discovered that the Miles's solution of the Garner equation with rectangular box initial data is not typical for the problem, as it captures only one soliton of large amplitude. A general smooth initial data captures however a number of solitons of large and small amplitudes. This simple idea is explained by the spectral AKNS problem, for which the class of smooth potentials lead to essentially different features compared to the class of continuous piecewise constant potentials. Download our paper published in Chaos.
• Spring-Summer, 2002: Functional analysis: rigorous proof of instability of standing waves of minimal energy. During my visit and collaborations with Andrew Comech (University of North Carolina, USA), we have rigorously proved nonlinear instability of standing waves of minimal energy in the framework of Grillakis-Shatah-Strauss formalism. As part of analysis, a "normal form" equation for standing waves of near-minimal energy is recovered, with rigorous bounds on error terms. The approach applies to the NLS systems, where the "normal form" equation was previously derived in my less rigorous publication with the use of asymptotic multi-scale expansion methods. Download our paper published in Communications in Pure and Applied Mathematics.
• Winter-Summer, 2002: Numerical analysis: proof of convergence of Petviashvili's iteration method. In collaborations with Yu. Stepanyants (ANSTO, Australia), we have developed the first analytical proof of convergence of the empiric Petviashvili's numerical method for finding stationary solutions of nonlinear wave equations. The proof is based on contraction mapping principle for nonlinear operators and spectral theory for linearized operators of nonlinear wave equations. We have confirmed Petviashvili's hypothesis on the fastest rate of convergence of the iteration method. Applications of the Petviashvili's method for KdV, BO, ZK, KP, and Klein-Gordon equations are discussed and numerical approximations of solitons and lumps in space of one and two dimensions are obtained. Download our paper published in SIAM Journal of Numerical Analysis.
• Winter-Spring, 2002: Semiconductors: computations of contact capacitance in Schottky depletion layer problem for non-uniformly doped semiconductors. Perturbation series expansions and variational approximation methods are developed in collaborations with Walter Craig (McMaster University) and Alex Shik (Centre for Advanced Nanotechnology, University of Toronto). The effective concentration is computed from the capacitance-voltage dependence of a semiconductor depletion layer, which is defined by a solution of nonlinear boundary-value elliptic problem. Elegant applications of Fourier series and exact integration of a variational system of differential equations are performed to simplify the Poisson equation with a zero-potential nonlinear surface and to study analytically the parameter dependence of the effective concentration. Download our paper published in J. Phys. D.: Appl. Phys.