Standing Waves with Periodic Amplitude in the Discrete Nonlinear Schrödinger Type Equation with Saturable Nonlinearity on 2D-Lattice

• Дата: 2018
• Автори: Бак, Сергій Миколайович, Ковтонюк, Галина Миколаївна, Печериця, Іван Володимирович
• Формат: Стаття
• Мова: Ukrainian
• Опубліковано: Кам'янець-Подільський національний університет імені Івана Огієнка 2018
• Онлайн доступ: http://mcm-math.kpnu.edu.ua/article/view/159248
• Назва журналу: Mathematical and computer modelling. Series: Physical and mathematical sciences

This paper is devoted to the study of a discrete nonlinear Schrödinger equation on a two-dimensional lattice. This type of equations with saturable nonlinearity is studied. We first consider the Schrödinger type equation with a more general nonlinearity, which has the same properties as saturable nonlinearity. For such equations, we obtain the result of the existence of solutions in the form of standing waves with periodic amplitude (note that such solutions are often called breathers). To do this, the given equation is presented by operator form in the space of two-sided sequences. It is assumed that the coefficients of the corresponding linear operator form k-periodic sequences. This operator is bounded and self-adjoint in the space of all k-periodic sequences. Then a special functional was constructed, the critical points of which in this space are solutions of the original equation. A Gateaux derivative of this functional is found. Next we consider the Nehari manifold for a given variational problem, which is a set of nontrivial critical points of a constructed functional in the space of k-periodic sequences. It is shown that this manifold is a non-empty and closed subset of a given space. In addition, the corresponding minimization problem is considered and it is shown that this problem has a solution in the Nehari manifold. Consequently, under these conditions the original equation has nontrivial periodic solutions. Finally, due to the fact that saturable nonlinearity satisfies these conditions, the existence of two nontrivial standing waves with k-periodic amplitude for a discrete nonlinear Schrödinger equation with saturable nonlinearity on a two-dimensional lattice is established. The results of this paper are the distribution of already known results for discrete nonlinear Schrödinger equations on 1D and 2D-lattices.