Journal of Mathematical Sciences, Vol. 174, No. 4, April, 2011 Traveling waves in systems of oscillators on 2D-lattices Sergiy N. Bak and Alexander A. Pankov Presented by A. E. Shishkov Abstract. A system of differential equations that describes the dynamics of an infinite system of linearly coupled nonlinear oscillators on a 2D-lattice is considered. The exponential estimate of the solution and some results on the existence of periodic and solitary traveling waves are obtained. Keywords. Oscillators, traveling waves, critical points, mountain pass theorem. 1. Introduction In the present work, we study the equations describing the dynamics of an infinite system of linearly coupled nonlinear oscillators positioned on a plane integer-valued lattice. Let q n,m (t) be the generalized coordinate of the (n, m)-th oscillator at the time t. It is assumed that each oscillator interacts linearly with its four nearest neighbors. The equations of motion of the system under consideration take the form ¨ q n,m = -U ′ (q n,m )+ c 2 1 (q n+1,m + q n-1,m - 2q n,m ) + c 2 2 (q n,m+1 + q n,m-1 - 2q n,m ), (n, m) ∈ Z 2 . (1.1) Equations (1.1) represent an infinite system of ordinary differential equations. Similar systems are of interest in connection with numerous physical applications [1, 4, 5]. In works [2,3,8], traveling waves in chains of oscillators were studied. The review of the available results concerning such systems is given in [11]. The periodic solutions for a system of oscillators on a two-dimensional lattice were studied in [14], and the traveling waves in similar systems of somewhat different types were considered in [6] and [7] within other methods. In particular, the system with odd 2π-periodic nonlinearity was analyzed in [6]. Here, we will study the question about the existence of periodic and solitary traveling waves within the method of critical points and will establish the exponential estimate of the profile of a traveling wave. 2. Statement of the problem Consider the system of oscillators with the potential U (r)= - a 2 r 2 + V (r). Then the equation of motion takes the form ¨ q n,m = c 2 1 △ (1) q n,m + c 2 2 △ (2) q n,m + aq n,m - V ′ (q n,m ), (2.1) Translated from Ukrains’ki˘ ı Matematychny˘ ı Visnyk, Vol. 7, No. 2, pp. 154–175, April–May, 2010. Original article submitted September 9, 2009 1072 – 3374/11/1744–0437 c  2011 Springer Science+Business Media, Inc. 437