Homoclinic traveling waves in discrete sine–Gordon equation with nonlinear interaction
on 2D–lattice

S. M. Bak

doi:10.30970/ms.52.2.176-184

The article deals with the discrete sine–Gordon equation that describes an infinite system of particles with nonlinear interaction and on-site potential on 2D–lattice. The main result concerns the existence of homoclinic traveling waves solutions. By means of critical point theory, we obtain sufficient conditions for the existence of such solutions.

1. S. Aubry, Breathers in nonlinear lattices: Existence, linear stability and quantization, (in Russian); Physica D, 103 (1997), 201–250.

2. S.M. Bak, Existence of heteroclinic traveling waves in a system of oscillators on a two-dimensional lattice, Matematychni Metody ta Fizyko-Mekhanichni Polya, 57 (2014), №3, 45–52. (in Ukrainian); Engl. transl.: J. Math. Sci., 217 (2016), №2, 187–197.

3. S.M. Bak, Existence of periodic traveling waves in Fermi-Pasta-Ulam system on 2D-lattice, Mat. Stud., 37 (2012), №1, 76–88. (in Ukrainian)

4. S.M. Bak, Existence of periodic traveling waves in systems of nonlinear oscillators on 2D-lattice, (in Russian); Mat. Stud., 35 (2011), №1, 60–65. (in Ukrainian)

5. S.M. Bak, Existence of the solitary traveling waves for a system of nonlinearly coupled oscillators on the 2d-lattice, Ukr. Mat. Zh., 69 (2017), №4, 435–444. (in Ukrainian) Engl. transl.: Ukr. Mat. J., 69 (2017), №4, 509–520.

6. S.M. Bak, Existence of the subsonic periodic traveling waves in the system of nonlinearly coupled nonlinear oscillators on 2D-lattice, Math. and Comp. Modelling. Ser.: Phys. and Math. Sci., 10 (2014), 17–23. (in Ukrainian)

7. S.M. Bak, Existence of the supersonic periodic traveling waves in the system of nonlinearly coupled nonlinear oscillators on 2D-lattice, Math. and Comp. Modelling. Ser.: Phys. and Math. Sci., 12 (2015), 5–12. (in Ukrainian)

8. S.M. Bak, Existence of the time periodic solutions of system of oscillators on 2D-lattice, Carpathian Mathematical Publications, 4 (2012), №2, 5–12. (in Ukrainian)

9. S.M. Bak, Peridoc traveling waves in chains of oscillators, Communications in Mathematical Analysis, 3 (2007), №1, 19-26.

10. S.M. Bak, Periodic traveling waves in the discrete sine.Gordon equation on 2D-lattice, Math. and Comp. Modelling. Ser.: Phys. and Math. Sci., 9 (2013), 5-10. (in Ukrainian)

11. S. Bak, The existence of heteroclinic traveling waves in the discrete sine-Gordon equation with nonlinear interaction on a 2D-lattice, J. Math. Phys., Anal., Geom., 14 (2018), №1, 16-26.

12. S.M. Bak, Travelling waves in chains of oscillators, Mat. Stud., 26 (2006), №2, 140-153. (in Ukrainian)

13. S.M. Bak, G. M. Kovtonyuk, Existence of solitary traveling waves in Fermi-Pasta-Ulam system on 2D lattice, Mat. Stud., 50 (2018), №1, 75-87.

14. S.N. Bak, A.A. Pankov, Traveling waves in systems of oscillators on 2D-lattices, Ukrainian Mathematical Bulletin, 7 (2010), №2, 154-175. (in Russian); Engl. transl.: J. Math. Sci., 174 (2011), №4, 916--920.

15. O.M. Braun, Y.S. Kivshar, Nonlinear dynamics of the Frenkel.Kontorova model, Physics Repts, 306 (1998), 1-108.

16. O.M. Braun, Y.S. Kivshar, The Frenkel-Kontorova Model, Concepts, Methods and Applications, Springer, Berlin, 2004.

17. M. Fe.ckan, V. Rothos, Traveling waves in Hamiltonian systems on 2D lattices with nearest neighbour interactions, (in Russian); Nonlinearity, 20 (2007), 319-341.

18. G. Friesecke, K. Matthies, Geometric solitary waves in a 2D math-spring lattice, Discrete and continuous dynamical systems, 3 (2003), №1, 105-114.

19. G. Ioos, K. Kirchgassner, Traveling waves in a chain of coupled nonlinear oscillators, Commun. Math. Phys., 211 (2000), 439-464.

20. C.F. Kreiner, J. Zimmer, Heteroclinic traveling waves for the lattice sine-Gordon equation with linear pair interaction, Discrete and continuous dynamical systems, 25 (2009), №3, 1-17.

21. C.F. Kreiner, J. Zimmer, Travelling wave solutions for the discrete sine-Gordon equation with nonlinear pair interaction, Nonlin. Analysis: Theory Methods and Appl., 70 (2009), №9, 3146-3158.

22. E.H. Lieb, On the lowest eigenvalue of the Laplacian for the intersection of two domains, Invent. Math., 74 (1983), №3, 441-448.

23. P.D. Makita, Periodic and homoclinic traveling waves in infinite lattices, Nonlin. Anal., 74 (2011), 2071-2086.

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

25. P. Srikanth, On periodic motions of two-dimentional lattices, Functional analysis with current applications in science, technology and industry, 377 (1998), 118-122.

26. M. Willem, Minimax theorems, Birkhauser, Boston, 1996.

	Homoclinic traveling waves in discrete sine–Gordon equation with nonlinear interaction on 2D–lattice
Author	S. M. Bak sergiy.bak@gmail.com Vinnytsia Mykhailo Kotsiubynskyi State Pedagogical University, Vinnytsia, Ukraine
Abstract	The article deals with the discrete sine–Gordon equation that describes an infinite system of particles with nonlinear interaction and on-site potential on 2D–lattice. The main result concerns the existence of homoclinic traveling waves solutions. By means of critical point theory, we obtain sufficient conditions for the existence of such solutions.
Keywords	discrete sine–Gordon equation; 2D–lattice; homoclinic traveling waves; critical points
DOI	doi:10.30970/ms.52.2.176-184
Reference	1. S. Aubry, Breathers in nonlinear lattices: Existence, linear stability and quantization, (in Russian); Physica D, 103 (1997), 201–250. 2. S.M. Bak, Existence of heteroclinic traveling waves in a system of oscillators on a two-dimensional lattice, Matematychni Metody ta Fizyko-Mekhanichni Polya, 57 (2014), №3, 45–52. (in Ukrainian); Engl. transl.: J. Math. Sci., 217 (2016), №2, 187–197. 3. S.M. Bak, Existence of periodic traveling waves in Fermi-Pasta-Ulam system on 2D-lattice, Mat. Stud., 37 (2012), №1, 76–88. (in Ukrainian) 4. S.M. Bak, Existence of periodic traveling waves in systems of nonlinear oscillators on 2D-lattice, (in Russian); Mat. Stud., 35 (2011), №1, 60–65. (in Ukrainian) 5. S.M. Bak, Existence of the solitary traveling waves for a system of nonlinearly coupled oscillators on the 2d-lattice, Ukr. Mat. Zh., 69 (2017), №4, 435–444. (in Ukrainian) Engl. transl.: Ukr. Mat. J., 69 (2017), №4, 509–520. 6. S.M. Bak, Existence of the subsonic periodic traveling waves in the system of nonlinearly coupled nonlinear oscillators on 2D-lattice, Math. and Comp. Modelling. Ser.: Phys. and Math. Sci., 10 (2014), 17–23. (in Ukrainian) 7. S.M. Bak, Existence of the supersonic periodic traveling waves in the system of nonlinearly coupled nonlinear oscillators on 2D-lattice, Math. and Comp. Modelling. Ser.: Phys. and Math. Sci., 12 (2015), 5–12. (in Ukrainian) 8. S.M. Bak, Existence of the time periodic solutions of system of oscillators on 2D-lattice, Carpathian Mathematical Publications, 4 (2012), №2, 5–12. (in Ukrainian) 9. S.M. Bak, Peridoc traveling waves in chains of oscillators, Communications in Mathematical Analysis, 3 (2007), №1, 19-26. 10. S.M. Bak, Periodic traveling waves in the discrete sine.Gordon equation on 2D-lattice, Math. and Comp. Modelling. Ser.: Phys. and Math. Sci., 9 (2013), 5-10. (in Ukrainian) 11. S. Bak, The existence of heteroclinic traveling waves in the discrete sine-Gordon equation with nonlinear interaction on a 2D-lattice, J. Math. Phys., Anal., Geom., 14 (2018), №1, 16-26. 12. S.M. Bak, Travelling waves in chains of oscillators, Mat. Stud., 26 (2006), №2, 140-153. (in Ukrainian) 13. S.M. Bak, G. M. Kovtonyuk, Existence of solitary traveling waves in Fermi-Pasta-Ulam system on 2D lattice, Mat. Stud., 50 (2018), №1, 75-87. 14. S.N. Bak, A.A. Pankov, Traveling waves in systems of oscillators on 2D-lattices, Ukrainian Mathematical Bulletin, 7 (2010), №2, 154-175. (in Russian); Engl. transl.: J. Math. Sci., 174 (2011), №4, 916--920. 15. O.M. Braun, Y.S. Kivshar, Nonlinear dynamics of the Frenkel.Kontorova model, Physics Repts, 306 (1998), 1-108. 16. O.M. Braun, Y.S. Kivshar, The Frenkel-Kontorova Model, Concepts, Methods and Applications, Springer, Berlin, 2004. 17. M. Fe.ckan, V. Rothos, Traveling waves in Hamiltonian systems on 2D lattices with nearest neighbour interactions, (in Russian); Nonlinearity, 20 (2007), 319-341. 18. G. Friesecke, K. Matthies, Geometric solitary waves in a 2D math-spring lattice, Discrete and continuous dynamical systems, 3 (2003), №1, 105-114. 19. G. Ioos, K. Kirchgassner, Traveling waves in a chain of coupled nonlinear oscillators, Commun. Math. Phys., 211 (2000), 439-464. 20. C.F. Kreiner, J. Zimmer, Heteroclinic traveling waves for the lattice sine-Gordon equation with linear pair interaction, Discrete and continuous dynamical systems, 25 (2009), №3, 1-17. 21. C.F. Kreiner, J. Zimmer, Travelling wave solutions for the discrete sine-Gordon equation with nonlinear pair interaction, Nonlin. Analysis: Theory Methods and Appl., 70 (2009), №9, 3146-3158. 22. E.H. Lieb, On the lowest eigenvalue of the Laplacian for the intersection of two domains, Invent. Math., 74 (1983), №3, 441-448. 23. P.D. Makita, Periodic and homoclinic traveling waves in infinite lattices, Nonlin. Anal., 74 (2011), 2071-2086. 24. A. Pankov, Traveling Waves and Periodic Oscillations in Fermi-Pasta-Ulam Lattices, Imperial College Press, London.Singapore, 2005. 25. P. Srikanth, On periodic motions of two-dimentional lattices, Functional analysis with current applications in science, technology and industry, 377 (1998), 118-122. 26. M. Willem, Minimax theorems, Birkhauser, Boston, 1996.
Pages	176-184
Volume	52
Issue	2
Year	2019
Journal	Matematychni Studii
Full text of paper	pdf
Table of content of issue	html

Homoclinic traveling waves in discrete sine–Gordon equation with nonlinear interaction on 2D–lattice