Periodic travelling waves in chains of oscillators  (in Ukrainian)

S.M.Bak

We consider a system of differential equations that describes the dynamics of an infinite chain of linearly coupled nonlinear oscillators. Results on existence of the periodic and solitary travelling waves and exponential decay of the solution are obtained.

1. Бак С.Н., Панков А.А. О периодических колебаниях бесконечной цепочки линейно связанных нелинейных осцилляторов // Доповіді НАН України. -- 2004. -- № 9.

2. Бак С.Н. Метод условной минимизации в задаче о колебаниях цепочки нелинейных осцилляторов // Математическая физика, анализ, геометрия. -- 2004. -- Т.11. -- № 3 . -- С. 263--273.

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

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

5. Braun O.M., Kivshar Y.S. The Frenkel--Kontorova model. -- Berlin: Springer, 2004. -- 427 pp.

6. Ioos G., Kirchgässner K. Traveling waves in a chain of coupled nonlinear oscillators // Commun. Math. Phys. -- 2000. -- V. 211, № 3. -- P. 439--464.

7. Lions P.L. The concentration -- compactness principle in the calculus of variation. The locally compact case I, II // Ann. Inst. H. Poincare, Anal. Non Lineaire. -- 1984. -- V. 1. -- P. 223 -- 238.

8. MacKay R.S., Aubry S. Proof of existence of breathers for time-reversible a Hamiltonian networks of weakly coupled oscillators // Nonlinearity. -- 1994. -- P. 1623--1643.

9. Pankov A. Traveling Waves and Periodic Oscillations in Fermi -- Pasta -- Ulam Lattices. -- London -- Singapore: Imperial College Press, 2005. -- 196 pp.

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

11. Willem M. Minimax theorems. -- Boston, Birkh\"auser. -- 1996. -- 162 pp.

	Periodic travelling waves in chains of oscillators (in Ukrainian)
Author	S.M.Bak Sergey.bak@online.com.ua Vinnytsia Mykhailo Kotsiubynskyi State Pedagogical University
Abstract	We consider a system of differential equations that describes the dynamics of an infinite chain of linearly coupled nonlinear oscillators. Results on existence of the periodic and solitary travelling waves and exponential decay of the solution are obtained.
Keywords	system of differential equation, infinite chain, linear coupled nonlinear oscillators
DOI	doi:10.30970/ms.26.2.140-153
Reference	1. Бак С.Н., Панков А.А. О периодических колебаниях бесконечной цепочки линейно связанных нелинейных осцилляторов // Доповіді НАН України. -- 2004. -- № 9. 2. Бак С.Н. Метод условной минимизации в задаче о колебаниях цепочки нелинейных осцилляторов // Математическая физика, анализ, геометрия. -- 2004. -- Т.11. -- № 3 . -- С. 263--273. 3. Aubry S. Breathers in nonlinear lattices: Existence,linear stability and quantization // Physica D. -- 1997. -- 103. -- P. 201--250. 4. Braun O.M., Kivshar Y.S. Nonlinear dynamics of the Frenkel--Kontorova model // Physics Repts. -- 1998. -- 306. -- P. 1--108. 5. Braun O.M., Kivshar Y.S. The Frenkel--Kontorova model. -- Berlin: Springer, 2004. -- 427 pp. 6. Ioos G., Kirchgässner K. Traveling waves in a chain of coupled nonlinear oscillators // Commun. Math. Phys. -- 2000. -- V. 211, № 3. -- P. 439--464. 7. Lions P.L. The concentration -- compactness principle in the calculus of variation. The locally compact case I, II // Ann. Inst. H. Poincare, Anal. Non Lineaire. -- 1984. -- V. 1. -- P. 223 -- 238. 8. MacKay R.S., Aubry S. Proof of existence of breathers for time-reversible a Hamiltonian networks of weakly coupled oscillators // Nonlinearity. -- 1994. -- P. 1623--1643. 9. Pankov A. Traveling Waves and Periodic Oscillations in Fermi -- Pasta -- Ulam Lattices. -- London -- Singapore: Imperial College Press, 2005. -- 196 pp. 10. Rabinowitz P. Minimax methods in critical point theory with applications to differential equations. -- Providence, R. I.: American Math. Soc. -- 1986. -- 100 pp. 11. Willem M. Minimax theorems. -- Boston, Birkh\"auser. -- 1996. -- 162 pp.
Pages	140-153
Volume	26
Issue	2
Year	2006
Journal	Matematychni Studii
Full text of paper	pdf
Table of content of issue	html