01.01.02 (Differential equations, dynamical systems, and optimal control)
Birth date:
17.08.1956
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.
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.
Biography
Graduated from Faculty of Computational Mathematics and Cybernetics of M. V. Lomonosov Moscow State University (Department of computational mathematics) in 1978. Ph.D. thesis was defended in 1982. A list of my works contains about 50 titles. Since 1986 I have been working at the Bogoliubov Laboratory of Theoretical Physics of the Joint Institute for Nuclear Research, currently in a group of mathematical physicists.
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.
P. E. Zhidkov, “On the Bari Basis Property of the Eigenfunction System of a Nonlinear Integro-Differential Equation”, Differ. Uravn., 38:9 (2002), 1183–1189; Differ. Equ., 38:9 (2002), 1260–1267
P. E. Zhidkov, “Riesz basis property of the system of eigenfunctions for a non-linear problem of Sturm–Liouville type”, Mat. Sb., 191:3 (2000), 43–52; Sb. Math., 191:3 (2000), 359–368
P. E. Zhidkov, “On the eigenvalue problem for the Sturm–Liouville operator with a potential that depends on the spectral parameter”, Differ. Uravn., 33:1 (1997), 39–47; Differ. Equ., 33:1 (1997), 38–46
4.
P. E. Zhidkov, “Completeness of systems of eigenfunctions for the Sturm–Liouville operator with potential depending on the spectral parameter and for one non-linear problem”, Mat. Sb., 188:7 (1997), 123–138; Sb. Math., 188:7 (1997), 1071–1084
P. E. Zhidkov, “Invariant measures generated by higher conservation laws for the Korteweg–de Vries equations”, Mat. Sb., 187:6 (1996), 21–40; Sb. Math., 187:6 (1996), 803–822
P. E. Zhidkov, V. Zh. Sakbaev, “On the existence of a countable set of solutions of a certain nonlinear boundary value problem”, Differ. Uravn., 31:4 (1995), 630–640; Differ. Equ., 31:4 (1995), 585–593
1994
7.
P. E. Zhidkov, V. Zh. Sakbaev, “On a nonlinear ordinary differential equation”, Mat. Zametki, 55:4 (1994), 25–34; Math. Notes, 55:4 (1994), 351–357
P. E. Zhidkov, “On the existence of a countable set of solutions in a problem of polaron theory”, Mat. Sb., 183:2 (1992), 102–111; Russian Acad. Sci. Sb. Math., 75:1 (1993), 247–255
P. E. Zhidkov, “On the existence of the solution of the Cauchy problem and the stability of kink solutions of the nonlinear Schrödinger equation”, Sibirsk. Mat. Zh., 33:2 (1992), 73–79; Siberian Math. J., 33:2 (1992), 239–246
P. E. Zhidkov, “The Cauchy problem for the generalized Korteweg–de Vries equation with periodic initial data”, Differ. Uravn., 26:5 (1990), 823–829; Differ. Equ., 26:5 (1990), 591–596
P. E. Zhidkov, “On the solvability of Cauchy problem and stability of some solutions to the nonlinear Schrödinger equation”, Matem. Mod., 1:10 (1989), 155–160
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