On the Boundedness of the Global Solution of Cauchy Problem for Infinite System of Nonlinear Oscillators on 2D-lattice

• Datum: 2019
• 1. Verfasser: Бак, Сергій Миколайович
• Format: Artikel
• Sprache: Ukrainisch
• Veröffentlicht: Кам'янець-Подільський національний університет імені Івана Огієнка 2019
• Online Zugang: http://mcm-math.kpnu.edu.ua/article/view/188960
• Назва журналу: Mathematical and computer modelling. Series: Physical and mathematical sciences

This article is devoted to the study of an infinite-dimensional Hamiltonian system, which describes an infinite system of linearly coupled nonlinear oscillators on a two-dimensional lattice. This system is a counteble system of ordinary differential equations. It is convenient to consider this system as a differential-operator equation in Hilbert space of two-way sequences. The Cauchy problem for such equations in this space is considered. Previously, it has been proven that under certain circumstances this problem has a unique local and global solution. Local resolution follows from the standard results for differential equations in Banach spaces. The basic conditions here are the spatial periodicity of coefficients of the operator of linear interaction of oscillators and the Cauchy continuity of nonlinearity (which is defined as a derivative of the on-site potential of the oscillator system). This system, in Hamilton view, was used to establish global resolution. Recall that from a physical point of view the Hamiltonian represents the total energy of the system, i.e. the sum of kinetic and potential energy. The basic conditions, in addition to those mentioned above, are the non-positivity of the operator of linearly interact between the oscillators and the half-boundary below their potentials. However, the question remains whether the global solution is bounded. In this article it is established that under the same conditions of existence of a global solution, if the linear interaction operator is non-positive and the on-site potential at infinity is infinitely large (uniformly over both spatial variables), or the linear interaction operator is negatively defined, then this solution is bounded to any initial data from a given sequence space. To prove this, the fact that the Hamiltonian of the system retains a constant value on the initial data was used