DEPARTMENT OF MECHANICS AND APPLIED MATHEMATICS
SOLITON DYNAMICS
DESCRIPTION
Solitons in contemporary understanding were first described by N.J. Zabusky and M.D. Kruskal in 1965 and they form now a paradigm in mathematical physics. Soliton is a solitary wave with finite energy and the necessary conditions of its existence include nonlinearity and dispersion. Soliton dynamics is one of the hot topics due to wide applications in hydrodynamics, electronics, solid mechanics, biophysics, optical fibers etc. When the classical, so called Korteweg–de Vries (KdV) soliton is based on quadratic nonlinearity and cubic dispersion, then the contemporary wave dynamics needs much more complicated mechanisms to be taken into account.
Earlier studies in the Institute of Cybernetics and Tallinn University of Technology have given good results in the numerical study and in the spectral analysis of the classical KdV equation and some of its modifications. The pseudospectral methods (used for numerical integration) and the idea of spectral analysis give additional information on the energetical background of wave structures beside their spatial-temporal profiles. This additional information is deeply related to the concept of solitons as the particle-like waves with a certain energy. During last years main attention have been paid to soliton formation and propagation mechanism in solids with microstructure. These problems are essential in crystalline solids where the dislocations or shape–memory effects are of importance. Such materials are now widely used in contemporary high technology. The soliton dynamics in materials with the higher order nonlinearity and higher order dispersion is one subject of our current studies. Namely, in the problem under consideration, the nonlinearity is described by the quartic elastic potential and dispersion by both the third and the fifth order derivatives. Particular problem is related to wave propagation in martensitic alloys. The second problem we consider is related to wave propagation in microstructured layers with energy influx. The last problem is mathematically described by the perturbed KdV equation, i.e., the influence of external force is described by a function in the r.h.s. of KdV equation. Two dimensional nonlinear wave motion, described by the Kadomtsev-Petviashvili equation, is the third current problem. Here the special interest is turned to soliton solutions which are analyzed in terms of Hirota bilinear formalism.
Triple-soliton formation and propagation from harmonic input in media with higher order dispersion and higher order nonlinearity.
STAFF
- Jüri Engelbrecht, D.Sc. Head of Division of Institute of Cybernetics, Professor of Tallinn University of Technology, Fellow of Estonian Academy of Sciences
- Andrus Salupere, Ph.D. (Cand. Sc.), senior research fellow of Institute of Cybernetics, Assistant Professor of Tallinn University of Technology
- Pearu Peterson, research fellow of Institute of Cybernetics, PhD student of Tallinn University of Technology
- Olari Ilison, student of Tallinn University of Technology
PARTNERS
- University of Paris 6 (prof. G.A. Maugin)
- Berlin Technical University (prof. W. Muschik)
- Aachen Technical University (prof. J. Ballmann)
ESF GRANTS
- No. 937, 1994
- No. 1487, 1995–1996
- No. 2631, 1997–1998 (expected to finish in 1999)
- BMBF Grant N 03N9005 2, 1996-1999
PUBLICATIONS in 1997
2 Research Reports; 1 Book; 4 Papers; 2 Abstracts; 6 talks at international conferences or seminars.
07/04/1998 webmaster