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

Author S. M. Bak
sergiy.bak@gmail.com
Âiííèöüêèé äåðæàâíèé ïåäàãîãi÷íèé óíiâåðñèòåò iì. Ì. Êîöþáèíñüêîãî

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. Áàê Ñ.Ì. Áiæó÷i õâèëi â ëàíöþãàõ îñöèëÿòîðiâ// Mat. Stud. – 2006. – Ò.26, ¹2. – Ñ. 140–153.

2. Áàê Ñ.Í., Ïàíêîâ À.À. Áåãóùèå âîëíû â ñèñòåìàõ îñöèëëÿòîðîâ íà äâóìåðíûõ ðåøåòêàõ// Óêðà¿íñüêèé ìàòåìàòè÷íèé âiñíèê. – 2010. – Ò.7, ¹2. – Ñ. 154–175.

3. Âàéíáåðã Ì.Ì. Âàðèàöèîííûé ìåòîä è ìåòîä ìîíîòîííûõ îïåðàòîðîâ. – Ì.: Íàóêà, 1972. – 415 ñ.

4. Êðàñíîñåëüñêèé Ì.À. Òîïîëîãè÷åñêèå ìåòîäû â òåîðèè íåëèíåéíûõ èíòåãðàëüíûõ óðàâíåíèé. – Ì.: Ãîñòåõèçäàò, 1956. – 392 ñ.

5. Ðèä Ì., Ñàéìîí Á. Ìåòîäû ñîâðåìåííîé ìàòåìàòè÷åñêîé ôèçèêè:  4-õ ò. – Ì.: Ìèð, 1978, Ò.2. – 395 ñ.

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, ¹1. – Ð. 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, ¹1. – 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, Birkh¨auser, 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