Periodic traveling waves in Fermi–Pasta–Ulam type systems with nonlocal interaction on 2d-lattice

S. M. Bak; G. M. Kovtonyuk

doi:10.30970/ms.60.2.180-190

S. M. Bak Vinnytsia Mykhailo Kotsiubynskyi State Pedagogical University Vinnytsia, Ukraine
G. M. Kovtonyuk Vinnytsia Mykhailo Kotsiubynskyi State Pedagogical University Vinnytsia, Ukraine

DOI: https://doi.org/10.30970/ms.60.2.180-190

Keywords: Fermi–Pasta–Ulam type systems;, nonlocal interaction;, periodic traveling waves;, 2D-lattice,, critical points;, Mountain Pass Theorem,, Cerami condition;, Nehari manifold.

Abstract

The paper deals with the Fermi--Pasta--Ulam type systems that describe an infinite systems of nonlinearly coupled particles with nonlocal interaction on a two dimensional lattice. It is assumed that each particle interacts nonlinearly with several neighbors horizontally and vertically on both sides. The main result concerns the existence of traveling waves solutions with periodic relative displacement profiles. We obtain sufficient conditions for the existence of such solutions with the aid of critical point method and a suitable version of the Mountain Pass Theorem for functionals satisfying the Cerami condition instead of the Palais--Smale condition. We prove that under natural assumptions there exist monotone traveling waves.

Author Biographies

S. M. Bak, Vinnytsia Mykhailo Kotsiubynskyi State Pedagogical University Vinnytsia, Ukraine

Vinnytsia Mykhailo Kotsiubynskyi State Pedagogical University
Vinnytsia, Ukraine

G. M. Kovtonyuk, Vinnytsia Mykhailo Kotsiubynskyi State Pedagogical University Vinnytsia, Ukraine

Vinnytsia Mykhailo Kotsiubynskyi State Pedagogical University
Vinnytsia, Ukraine

References

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

S.M. Bak, Discrete infinite-dimensional Hamiltonian systems on a two-dimensional lattice, Doctor’s thesis, VSPU, Vinnytsia, 2020.

S.M. Bak, Existence of the solitary traveling waves for a system of nonlinearly coupled oscillators on the 2d-lattice, Ukr. Mat. J., 69 (2017), №4, 509–520.

S. Bak, Periodic traveling waves in the system of linearly coupled nonlinear oscillators on 2D lattice, Archivum Mathematicum, 58 (2022), №1, 1–13.

S. Bak, Periodic traveling waves in a system of nonlinearly coupled nonlinear oscillators on a twodimensional lattice, Acta Mathematica Universitatis Comenianae, 91 (2022), №3, 225–234.

S.M. Bak, G.M. Kovtonyuk, Existence of periodic traveling waves in Fermi–Pasta–Ulam type systems on 2D-lattice with saturable nonlinearities, J. Math. Sci., 260 (2022), №5, 619–629.

S.M. Bak, G.M. Kovtonyuk, Existence of traveling solitary waves in Fermi–Pasta–Ulam type systems on 2D-lattice with saturable nonlinearities, J. Math. Sci., 270 (2023), №3, 397–406.

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.

S.M. Bak, G.M. Kovtonyuk, Existence of traveling waves in Fermi–Pasta–Ulam type systems on 2D–lattice, J. Math. Sci., 252 (2020), №4, 453–462.

S.N. Bak, A.A. Pankov, Traveling waves in systems of oscillators on 2D-lattices, J. Math. Sci., 174 (2011), №4, 916–920.

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

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

I.A. Butt, J.A.D. Wattis, Discrete breathers in a two-dimensional Fermi–Pasta–Ulam lattice, J. Phys. A. Math. Gen., 39 (2006), 4955–4984.

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

G. Friesecke, K. Matthies, Geometric solitary waves in a 2D math-spring lattice, Discrete Contin. Dyn. Syst., 3 (2003), №1, 105–114.

D. Henning, G. Tsironis, Wave transmission in nonliniear lattices, Physics Repts., 309 (1999), 333–432.

D. Motreanu, V. Motreanu, N. Papageorgiou, Topological and variational methods with applications to boundary value problems, Springer, New York, 2014.

A. Pankov, Traveling waves in Fermi–Pasta–Ulam chains with nonlocal interaction, Discrete Contin. Dyn. Syst., 12 (2019), №7, 2097–2113.

A. Pankov, Traveling waves and periodic oscillations in Fermi-Pasta-Ulam lattices, Imperial College Press, London—Singapore, 2005.

P. Rabinowitz, Minimax methods in critical point theory with applications to differential equations, American Math. Soc., Providence, R. I., 1986.

J.A.D. Wattis, Approximations to solitary waves on lattices: III. The monoatomic lattice with secondneighbour interaction, J. Phys. A: Math. Gen., 29 (1996), 8139–8157.

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