Existence of standing waves in discrete nonlinear equation of Schrödinger type with saturable nonlinearity

S.M.Bak

In this paper we obtained results on existence of standing waves in discrete nonlinear equation of Schrödinger type with saturable nonlinearity. We consider two types of solutions: with periodic amplitude and vanishing at infinity. Calculus of variations and Nehari manifold are employed to establish the existence of these solutions.

1. Бак С. Н., Панков А. А. О периодических колебаниях бесконечной цепочки линейно связанных нелинейных осцилляторов // Доповіді НАН України. - 2004. - №9. - C. 13-16.

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

3. Бак С. М. Біжучі хвилі в ланцюгах осциляторів // Математичні студії. - 2006. - Т. 26, №2. - С.~140-153.

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

5. Bak S. M. Peridoc traveling waves in chains of oscillators// Communications in Mathematical Analysis. - 2007. - V. 3, N 1. - Р. 19-26.

6. Henning D., Tsironis G. Wave transmission in nonliniear lattices // Physics Repts. - 1999. - V. 309. - P. 333-432.

7. Pankov A. Traveling waves and periodic oscillations in Fermi-Pasta-Ulam Lattices. - London-Singapore: Imperial College Press, 2005. - 196 pp.

8. Pankov A. Gap solitons in periodic discrete NLS equations// Nonlinearity. - 2006. - V. 19. - P. 27-40.

9. Pankov A. Gap solitons in periodic discrete NLS equations II: A generalized Nehari manifold approach// Discr. Cont. Dyn. Syst. - 2007. - V. 19. - P. 419-430.

10. Pankov A., Rothos V. Periodic and decaying solutions in DNLS with saturable nonlinearity// Proc. Roy. Soc. A. - To appear.

11. Teschl G. Jacobi operators and completely integrable nonlinear lattices, Amer. Math. Soc., Providence, 2000. - 251 pp.

	Existence of standing waves in discrete nonlinear equation of Schrödinger type with saturable nonlinearity (in Ukrainian)
Author	S.M.Bak sergiy.bak@gmail.com Vinnytsia Mykhailo Kotsiubynskyi State Pedagogical University
Abstract	In this paper we obtained results on existence of standing waves in discrete nonlinear equation of Schrödinger type with saturable nonlinearity. We consider two types of solutions: with periodic amplitude and vanishing at infinity. Calculus of variations and Nehari manifold are employed to establish the existence of these solutions.
Keywords	standing wave, discrete nonlinear equation, saturable nonlinearity, periodic amplitude, Nehari manifold
DOI	doi:10.30970/ms.33.1.78-84
Reference	1. Бак С. Н., Панков А. А. О периодических колебаниях бесконечной цепочки линейно связанных нелинейных осцилляторов // Доповіді НАН України. - 2004. - №9. - C. 13-16. 2. Бак С. Н. Метод условной минимизации в задаче о колебаниях цепочки нелинейных осцилляторов //Математическая физика, анализ, геометрия. - 2004. - №3. - Т.11. - С. 263-273. 3. Бак С. М. Біжучі хвилі в ланцюгах осциляторів // Математичні студії. - 2006. - Т. 26, №2. - С.~140-153. 4. Aubry S. Breathers in nonlinear lattices: Existence, linear stability and quantization // Physica D. - 1997. - V. 103. - P. 201 - 250. 5. Bak S. M. Peridoc traveling waves in chains of oscillators// Communications in Mathematical Analysis. - 2007. - V. 3, N 1. - Р. 19-26. 6. Henning D., Tsironis G. Wave transmission in nonliniear lattices // Physics Repts. - 1999. - V. 309. - P. 333-432. 7. Pankov A. Traveling waves and periodic oscillations in Fermi-Pasta-Ulam Lattices. - London-Singapore: Imperial College Press, 2005. - 196 pp. 8. Pankov A. Gap solitons in periodic discrete NLS equations// Nonlinearity. - 2006. - V. 19. - P. 27-40. 9. Pankov A. Gap solitons in periodic discrete NLS equations II: A generalized Nehari manifold approach// Discr. Cont. Dyn. Syst. - 2007. - V. 19. - P. 419-430. 10. Pankov A., Rothos V. Periodic and decaying solutions in DNLS with saturable nonlinearity// Proc. Roy. Soc. A. - To appear. 11. Teschl G. Jacobi operators and completely integrable nonlinear lattices, Amer. Math. Soc., Providence, 2000. - 251 pp.
Pages	78-84
Volume	33
Issue	1
Year	2010
Journal	Matematychni Studii
Full text of paper	pdf
Table of content of issue	html

Existence of standing waves in discrete nonlinear equation of Schrödinger type with saturable nonlinearity (in Ukrainian)