Existence of solitary traveling waves in Fermi-Pasta-Ulam system
on 2D–lattice

S. M. Bak; G. M. Kovtonyuk

doi:10.15330/ms.50.1.75-87

The article deals with the Fermi–Pasta–Ulam system that describes an infinite system of particles on 2D–lattice. The main result concerns the existence of solitary traveling wave 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, 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. in 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, 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. in 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, Mat. 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, Mat. 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, Mat. and Comp. Modelling. Ser.: Phys. and Math. Sci., 9 (2013), 5–10. (in Ukrainian)

11. S.M. Bak, The existence and uniqueness of the global solution of the Cauchy problem for an infinite system of nonlinear oscillators on a two-dimensional lattice, Mat. and Comp. Modelling. Ser.: Phys. and Math. Sci., 5 (2011), 3–9. (in Ukrainian)

12. 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.

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

14. S.M. Bak, O.O. Baranova, Yu.P. Bilyk, Well-posedness of the Cauchy problem for an infinite system of nonlinear oscillators on 2D–lattice, Mat. and Comp. Modelling. Ser.: Phys. and Math. Sci., 4 (2010), 18–24. (in Ukrainian)

15. S.N. Bak, A.A. Pankov, Traveling waves in systems of oscillators on 2D–lattices, Ukrains’kyi Matematychnyi Visnyk, 7 (2010), №2, 154–175. (in Russian) Engl. transl. in J. Math. Sci., 174 (2011), №4, 916–920.

16. S.M. Bak, K.Ye. Rumyantceva, Correctness of the Cauchy problem for an infinite system of nonlinear oscillators with cubic potential on a two-dimensional lattice, Mat. and Comp. Modelling. Ser.: Phys. and Math. Sci., 6 (2012), 29–36. (in Ukrainian)

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

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

19. M. Feckan, V. Rothos, Traveling waves in Hamiltonian systems on 2D lattices with nearest neighbour interactions, Nonlinearity, 20 (2007), 319-341.

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

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

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

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

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

	Existence of solitary traveling waves in Fermi-Pasta-Ulam system on 2D–lattice
Author	S. M. Bak¹, G. M. Kovtonyuk² sergiy.bak@gmail.com¹, galyna.kovtonyuk@gmail.com² 1) Vinnytsia Mykhailo Kotsiubynskyi State Pedagogical University, Vinnitsa, Ukraine; 2) Vinnitsa State Pedagogical University, Vinnitsa, Ukraine
Abstract	The article deals with the Fermi–Pasta–Ulam system that describes an infinite system of particles on 2D–lattice. The main result concerns the existence of solitary traveling wave solutions. By means of critical point theory, we obtain sufficient conditions for the existence of such solutions.
Keywords	Fermi-Pasta-Ulam system; 2D–lattice; solitary traveling waves; critical points
DOI	doi:10.15330/ms.50.1.75-87
Reference	1. S. Aubry, Breathers in nonlinear lattices: Existence, linear stability and quantization, 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. in 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, 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. in 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, Mat. 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, Mat. 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, Mat. and Comp. Modelling. Ser.: Phys. and Math. Sci., 9 (2013), 5–10. (in Ukrainian) 11. S.M. Bak, The existence and uniqueness of the global solution of the Cauchy problem for an infinite system of nonlinear oscillators on a two-dimensional lattice, Mat. and Comp. Modelling. Ser.: Phys. and Math. Sci., 5 (2011), 3–9. (in Ukrainian) 12. 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. 13. S.M. Bak, Travelling waves in chains of oscillators, Mat. Stud., 26 (2006), №2, 140–153. (in Ukrainian) 14. S.M. Bak, O.O. Baranova, Yu.P. Bilyk, Well-posedness of the Cauchy problem for an infinite system of nonlinear oscillators on 2D–lattice, Mat. and Comp. Modelling. Ser.: Phys. and Math. Sci., 4 (2010), 18–24. (in Ukrainian) 15. S.N. Bak, A.A. Pankov, Traveling waves in systems of oscillators on 2D–lattices, Ukrains’kyi Matematychnyi Visnyk, 7 (2010), №2, 154–175. (in Russian) Engl. transl. in J. Math. Sci., 174 (2011), №4, 916–920. 16. S.M. Bak, K.Ye. Rumyantceva, Correctness of the Cauchy problem for an infinite system of nonlinear oscillators with cubic potential on a two-dimensional lattice, Mat. and Comp. Modelling. Ser.: Phys. and Math. Sci., 6 (2012), 29–36. (in Ukrainian) 17. O.M. Braun, Y.S. Kivshar, Nonlinear dynamics of the Frenkel–Kontorova model, Physics Repts, 306 (1998), 1–108. 18. O.M. Braun, Y.S. Kivshar, The Frenkel-Kontorova Model, Concepts, Methods and Applications, Springer, Berlin, 2004. 19. M. Feckan, V. Rothos, Traveling waves in Hamiltonian systems on 2D lattices with nearest neighbour interactions, Nonlinearity, 20 (2007), 319-341. 20. G. Friesecke, K. Matthies, Geometric solitary waves in a 2D math-spring lattice, Discrete and continuous dynamical systems, 3 (2003), №1, 105-114. 21. G. Ioos, K. Kirchgassner, Traveling waves in a chain of coupled nonlinear oscillators, Commun. Math. Phys., 211 (2000), 439-464. 22. P.D. Makita, Periodic and homoclinic traveling waves in infinite lattices, Nonlin. Anal., 74 (2011), 2071-2086. 23. A. Pankov, Traveling Waves and Periodic Oscillations in Fermi-Pasta-Ulam Lattices, Imperial College Press, London-Singapore, 2005. 24. P. Srikanth, On periodic motions of two-dimentional lattices, Functional analysis with current applications in science, technology and industry, 377 (1998), 118-122.
Pages	75-87
Volume	50
Issue	1
Year	2018
Journal	Matematychni Studii
Full text of paper	pdf
Table of content of issue	html

Existence of solitary traveling waves in Fermi-Pasta-Ulam system on 2D–lattice