Geodesic
The existence of heteroclinic travelling waves in the discrete sine-Gordon equation with nonlinear interaction on a $\mathrm{2D}$-lattice
Žurnal matematičeskoj fiziki, analiza, geometrii, Tome 14 (2018) no. 1, pp. 16-26 Cet article a éte moissonné depuis la source Math-Net.Ru

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The article deals with the discrete sine-Gordon equation that describes an infinite system of nonlinearly coupled nonlinear oscillators on a $\mathrm{2D}$-lattice with the external potential $V(r)=K(1-\cos r)$. The main result concerns the existence of heteroclinic travelling waves solutions. Sufficient conditions for the existence of these solutions are obtained by using the critical points method and concentration-compactness principle. 
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S. Bak. The existence of heteroclinic travelling waves in the discrete sine-Gordon equation with nonlinear interaction on a $\mathrm{2D}$-lattice. Žurnal matematičeskoj fiziki, analiza, geometrii, Tome 14 (2018) no. 1, pp. 16-26. http://geodesic.mathdoc.fr/item/JMAG_2018_14_1_a1/

[1] S. M. Bak, “Traveling waves in chains of oscillators”, Mat. Stud., 26 (2006), 140–153 (Ukrainian) | MR | Zbl

[2] S. M. Bak, “Periodic traveling waves in chains of oscillators”, Commun. Math. Anal., 3 (2007), 19–26 | MR | Zbl

[3] 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) | MR | Zbl

[4] 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) | MR | Zbl

[5] S. M. Bak, “Periodic traveling waves in the discrete sine-Gordon equation on 2D-lattice”, Mat. Komp. Model. Ser.: Fiz.-Mat. Nauky, 9 (2013), 5–10 (Ukrainian)

[6] 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)

[7] 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) | Zbl

[8] J. Math. Sci. (N.Y.), 217 (2016) | DOI | MR

[9] Ukrainian Math. J., 69 (2017) | DOI | MR | Zbl

[10] J. Math. Sci. (N.Y.), 174 (2011) | DOI | Zbl

[11] O. M. Braun, Y. S. Kivshar, The Frenkel–Kontorova Model. Concepts, Methods, and Applications, Texts and Monographs in Physics, Springer-Verlag, Berlin, 2004 | MR | Zbl

[12] M. Fečkan, V. Rothos, “Travelling waves in Hamiltonian systems on 2D lattices with nearest neighbour interactions”, Nonlinearity, 20 (2007), 319–341 | DOI | MR

[13] G. Friesecke, K. Matthies, “Geometric solitary waves in a 2D math-spring lattice”, Discrete Contin. Dyn. Syst. Ser. B, 3 (2003), 105–114 | DOI | MR | Zbl

[14] G. Ioos, K. Kirchgässner, “Travelling waves in a chain of coupled nonlinear oscillators”, Comm. Math. Phys., 211 (2000), 439–464 | DOI | MR | Zbl

[15] C.-F. Kreiner, J. Zimmer, “Heteroclinic travelling waves for the lattice sine-Gordon equation with linear pair interaction”, Discrete Contin. Dyn. Syst., 25 (2009), 915–931 | DOI | MR | Zbl

[16] C.-F. Kreiner, J. Zimmer, “Travelling wave solutions for the discrete sine-Gordon equation with nonlinear pair interaction”, Nonlinear Anal., 70 (2009), 3146–3158 | DOI | MR | Zbl

[17] 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 | DOI | MR | Zbl

[18] P. D. Makita, “Periodic and homoclinic travelling waves in infinite lattices”, Nonlinear Anal., 74 (2011), 2071–2086 | DOI | MR | Zbl

[19] A. Pankov, Travelling Waves and Periodic Oscillations in Fermi–Pasta–Ulam Lattices, Imperial College Press, London, 2005 | MR