Existence of periodic travelling waves in systems of nonlinear oscillators on 2Dlattice (in Ukrainian)

Author S. M. Bak
sergiy.bak@gmail.com
Вiнницький державний педагогiчний унiверситет iм. М. Коцюбинського

Abstract It is considered the system of differential equations that describes the dynamics of an infinite system of linearly coupled nonlinear oscillators on 2D-lattice. Results on existence of the periodic travelling waves are obtained.
Keywords nonlinear oscillator; periodic travelling wave
Reference 1. Бак С.М. Бiжучi хвилi в ланцюгах осциляторiв// Mat. Stud. – 2006. – Т.26, №2. – С. 140–153.

2. Бак С.Н., Панков А.А. Бегущие волны в системах осцилляторов на двумерных решетках// Український математичний вiсник. – 2010. – Т.7, №2. – С. 154–175.

3. Вайнберг М.М. Вариационный метод и метод монотонных операторов. – М.: Наука, 1972. – 415 с.

4. Красносельский М.А. Топологические методы в теории нелинейных интегральных уравнений. – М.: Гостехиздат, 1956. – 392 с.

5. Рид М., Саймон Б. Методы современной математической физики: В 4-х т. – М.: Мир, 1978, Т.2. – 395 с.

6. Aubry S. Breathers in nonlinear lattices: Existence, linear stability and quantization// Physica D. – 1997. – V.103. – P. 201–250.

7. Bak S.M. Peridoc traveling waves in chains of oscillators// Communications in Mathematical Analysis. – 2007. – V.3, №1. – Р. 19–26.

8. Braun O.M., Kivshar Y.S. Nonlinear dynamics of the Frenkel–Kontorova model// Physics Repts. – 1998. – V.306. – P. 1–108.

9. Braun O.M., Kivshar Y.S. The Frenkel–Kontorova model. – Berlin: Springer, 2004. – 427 p.

10. Feckan M., Rothos V. Traveling waves in Hamiltonian systems on 2D lattices with nearest neighbour interactions// Nonlinearity. – 2007. – V.20. – P. 319–341.

11. Friesecke G., Matthies K. Geometric solitary waves in a 2D math-spring lattice// Discrete and continuous dynamical systems. – 2003. – V.3, №1. – P. 105–114.

12. Iooss G., Kirchgassner K. Traveling waves in a chain of coupled nonlinear oscillators// Commun. Math. Phys. – 2000. – V.211. – P. 439–464.

13. Pankov A. Periodic nonlinear Schrodinger equation with an application to photonic crystals// Milan J. Math. – 2005. – V.73. – P. 259–287.

14. Pankov A. Traveling waves and periodic oscillations in Fermi-Pasta-Ulam lattices. - London–Singapoore: Imperial College Press, 2005. – 196 p.

15. Rabinowitz P. Minimax methods in critical point theory with applications to differential equations. – Providence, R. I.: American Math. Soc., 1986. – 100 p.

16. Srikanth P. On periodic motions of two-dimentional lattices// Functional analysis with current applications in science, technology and industry. – 1998. – V.377. – P. 118–122.

17. Willem M. Minimax theorems. - Boston, BirkhЁauser, 1996. – 162 p.

Pages 60-65
Volume 35
Issue 1
Year 2011
Journal Matematychni Studii
Full text of paper PDF
Table of content of issue HTML