Speciality:
01.01.02 (Differential equations, dynamical systems, and optimal control)
Birth date:
17.08.1956
Phone: +7 (09621) 6 50 84, +7 (09621) 4 03 64
Fax: +7 (09621) 6 50 84
E-mail: Keywords: qualitative theory of nonlinear differential equations; stability of solitary waves; nonlinear second-order elliptic boundary-value problems; infinite-dimensional dynamical systems generated by nonlinear partial differential equations, constructing invariant measures for these systems; nonlinear spectral problems, analysis of basis properties for systems of their eigenfunctions.
UDC: 517.956, 517.957, 517.958, 517.987.4, 517.927.25 MSC: 34B15, 34L10, 34L30, 35B35, 35B38, 35J65, 35L70, 35Q51, 35Q53, 35Q55, 37K99, 46G12, 47J10, 47J35, 58E05
Subject:
A stability of a soliton solution of the cubic nonlinear Schroedinger equation is defined and proved. A stability of solitary waves that do not vanish as the spatial variable tends to infinity for similar equations of a more general kind is studied. The problem of constructing invariant measures associated with energy for the nonlinear Schroedinger equation is solved partially. Invariant measures associated with higher conservation laws are constructed for the Korteweg–de Vries and nonlinear Schroedinger equations. The property of being a basis in $L_2$ is proved for eigenfunctions of simplest nonlinear Sturm–Liouville-type problems. In addition, the well-posedness of a problem for the Vlasov equation with a regular potential, when one has a joint distribution of particles in a coordinate space, is shown and (with V. Zh. Sakbaev) questions of the existence of radial solutions for superlinear elliptic second-order boundary-value problems in a spherical layer in the case when coefficients depending only on the spatial variable in the equation change sign and questions of constructing of wave multifunctions in multiply connected domains are considered.
Main publications:
P. E. Zhidkov. An invariant measure for a nonlinear wave equation // Nonlinear Anal.: Theory, Meth. Appl. 1994. V. 22, no. 3. P. 319–325.
P. E. Zhidkov. Korteweg–de Vries and nonlinear Schroedinger equations: qualitative theory, Springer-Verlag, Heidelberg, 2001. (Lecture Notes in Mathematics. V. 1756.)
P. E. Zhidkov. On a problem with two-time data for the Vlasov equation // Nonlinear Anal.: Theory, Meth. Appl. 1998. V. 31, no. 6. P. 537–547.
