The Existence of Heteroclinic Travelling Waves in the Discrete Sine-Gordon Equation with Nonlinear Interaction on a 2D-Lattice
DOI:
https://doi.org/10.15407/mag14.01.016Анотація
Статтю присвячено дискретному рiвнянню синус-Ґордона, яке описує нескiнченну систему нелiнiйно зв’язаних нелiнiйних осциляторiв на двовимiрнiй ґратцi iз зовнiшнiм потенцiалом V (r) = K(1 - cos r). Основний результат стосується iснування розв’язкiв у виглядi гетероклiнiчних рухомих хвиль. За допомогою методу критичних точок i принципу концентрованої компактностi отримано достатнi умови iснування таких розв’язкiв.
Mathematical Subject Classification: 34G20, 37K60, 58E50.
Ключові слова:
дискретне рiвняння синус-Ґордона, нелiнiйнi осцилятори, двовимiрна ґратка, гетероклiнiчнi рухомi хвилi, критичнi точки, принцип концентрованої компактностiПосилання
S.M. Bak, Traveling waves in chains of oscillators, Mat. Stud. 26 (2006), 140–153 (Ukrainian).
S.M. Bak, Periodic traveling waves in chains of oscillators, Commun. Math. Anal. 3 (2007), 19–26.
S.M. Bak, Existence of periodic traveling waves in a system of nonlinear oscillators on a two-dimensional lattice, Mat. Stud. 35 (2011), 60–65 (Ukrainian).
S.M. Bak, Existence of periodic traveling waves in the Fermi–Pasta–Ulam system on a two-dimensional lattice, Mat. Stud. 37 (2012), 76–88 (Ukrainian).
S.M. Bak, Periodic traveling waves in the discrete sine-Gordon equation on 2Dlattice, Mat. Komp. Model. Ser.: Fiz.-Mat. Nauky 9 (2013), 5–10 (Ukrainian).
S.M. Bak, Existence of the subsonic periodic traveling waves in the system of nonlinearly coupled nonlinear oscillators on 2D-lattice, Mat. Komp. Model. Ser.: Fiz.-Mat. Nauky 10 (2014), 17–23 (Ukrainian).
S.M. Bak, Existence of the supersonic periodic traveling waves in the system of nonlinearly coupled nonlinear oscillators on 2D-lattice, Mat. Komp. Model. Ser.: Fiz.-Mat. Nauky 12 (2015), 5–12 (Ukrainian).
S.M. Bak, Existence of heteroclinic traveling waves in a system of oscillators on a two-dimensional lattice, Mat. Metodi Fiz.-Mekh. Polya 57 (2014), 45–52 (Ukrainian); Engl. transl.: J. Math. Sci. (N.Y.) 217 (2016), 187–197.
S.N. Bak, Existence of solitary traveling waves for a system of nonlinear coupled oscillators on a two-dimensional lattice, Ukraı̈n. Mat. Zh. 69 (2017), 435–444 (Ukrainian); Engl. transl.: Ukrainian Math. J. 69 (2017), 509–520.
S.N. Bak and A.A. Pankov, Traveling waves in systems of oscillators on twodimensional lattices, Ukr. Mat. Visn. 7 (2010), 154–175 (Ukrainian); Engl. transl.: J. Math. Sci. (N.Y.) 174 (2011), 437–452.
O.M. Braun and Y.S. Kivshar, The Frenkel–Kontorova Model. Concepts, Methods, and Applications. Texts and Monographs in Physics, Springer–Verlag, Berlin, 2004.
M. Fečkan and V. Rothos, Travelling waves in Hamiltonian systems on 2D lattices with nearest neighbour interactions, Nonlinearity 20 (2007), 319–341. https://doi.org/10.1088/0951-7715/20/2/005
G. Friesecke and K. Matthies, Geometric solitary waves in a 2D math-spring lattice, Discrete Contin. Dyn. Syst. Ser. B 3 (2003), 105–114. https://doi.org/10.3934/dcdsb.2003.3.105
G. Ioos and K. Kirchgässner, Travelling waves in a chain of coupled nonlinear oscillators, Comm. Math. Phys. 211 (2000), 439–464. https://doi.org/10.1007/s002200050821
C.-F. Kreiner and J. Zimmer, Heteroclinic travelling waves for the lattice sineGordon equation with linear pair interaction, Discrete Contin. Dyn. Syst. 25 (2009), 915–931. https://doi.org/10.3934/dcds.2009.25.915
C.-F. Kreiner and J. Zimmer, Travelling wave solutions for the discrete sine-Gordon equation with nonlinear pair interaction, Nonlinear Anal. 70 (2009), 3146–3158. https://doi.org/10.1016/j.na.2008.04.018
P.-L. Lions, The concentration–compactness principle in the calculus of variations. The locally compact case, I, II, Ann. Inst. H. Poincaré Anal. Non Linéaire 1 (1984), 223–283. https://doi.org/10.1016/S0294-1449(16)30422-X
P.D. Makita, Periodic and homoclinic travelling waves in infinite lattices, Nonlinear Anal. 74 (2011), 2071–2086. https://doi.org/10.1016/j.na.2010.11.011
A. Pankov, Travelling Waves and Periodic Oscillations in Fermi–Pasta–Ulam Lattices. Imperial College Press, London, 2005. https://doi.org/10.1142/p381
