Standing Waves in Discrete Klein-Gordon Type Equations with Saturable Nonlinearities

• Дата: 2021
• Автор: Бак, Сергій
• Формат: Стаття
• Мова: Українська
• Опубліковано: Кам'янець-Подільський національний університет імені Івана Огієнка 2021
• Онлайн доступ: http://mcm-math.kpnu.edu.ua/article/view/251135
• Назва журналу: Mathematical and computer modelling. Series: Physical and mathematical sciences

This article is devoted to the study of discrete Klein-Gordon type equations that describe the dynamics of infinite chains of linearly coupled nonlinear oscillators. Such equations are infinite systems of ordinary differential equations. Equations of this type with saturable nonlinearities are studied. For such equations, results on the existence of solutions in the form of standing waves are obtained. After substituting the ansatz in the form of a standing wave into this system, a system of algebraic equations for the amplitude of a standing wave is obtained. Two types of solutions are studied: periodic (with a period k) and localized (converging to zero at infinity). These equations have a variational structure. Therefore, it is shown that k-periodic and localized solutions can be constructed as critical points of some two functionals in the corresponding spaces of two-sided sequences. Next, we consider the Nehari manifolds for the corresponding variational problems. These manifolds contain nontrivial critical points of these functionals. It is shown that the Nehari manifolds are non-empty and closed submanifolds of the corresponding spaces of two-sided sequences. In addition, the corresponding problems of minimizing these functionals are considered. It is shown that on the Nehari manifold for the first functional the corresponding minimization problem has a solution under certain conditions. Therefore, under these conditions, the original equation has nontrivial k-periodic solutions. In the case of localized solutions, it is difficult to prove that the corresponding minimization problem has a solution on the corresponding Nehari manifold. Therefore, in this case, the method of periodic approximations is used, i.e., the critical points of the functional that corresponds to localized solutions are constructed using the passage to the limit (with a period k tending to infinity) at the critical points of the functional that corresponds to k-periodic solutions. The obtained localized solutions are the solutions of the corresponding minimization problem.