A NONLOCAL PROBLEM FOR A HYPERBOLIC EQUATION WITH A DOMINANT MIXED DERIVATIVE

In this article, we consider the Goursat problem with nonlocal integral conditions for a hyperbolic equation with a dominant mixed derivative. Research methods of solvability of classical boundary value problems for partial differential equations cannot be applied without serious modifications. The choice of a research method of solvability of a nonlocal problem depends on the form of the integral condition. In the process of developing methods that are effective for nonlocal problems, integral conditions of various types were identified [1]. The solvability of the nonlocal Goursat problem with integral conditions of the first kind for a general equation with dominant mixed derivative of the second order was investigated in [2]. In our problem, the integral conditions are nonlocal conditions of the second kind, therefore, to investigate the solvability of the problem, we propose another method, which consists in reducing the stated nonlocal problem to the classical Goursat problem, but for a loaded equation. In this article, we obtain conditions that guarantee the existence of a unique solution of the problem. The main instrument of the proof is the a priori estimates obtained in the paper.

Breathers and double-pole solutions of coupled mixed derivative nonlinear Schrödinger equations from optical fibers

In this paper, the coupled mixed derivative nonlinear Schrödinger equations are investigated, which govern the propagation of the femtosecond optical pulse in optical fibers. First of all, based on the soliton solutions in bilinear form, the breathers are constructed by choosing a pair of complex conjugate wave numbers. Then, the interactions between a breather and either an anti-dark soliton or a dark soliton are studied according to the existence conditions of dark and anti-dark solitons. The double-pole solution can also be obtained by a coalescence of two wave numbers. In addition, the influence of physical parameters on the phases and propagation direction of the breathers and double-pole solitons is studied by the qualitative analysis and graphical illustration.

Equivalent Characterization on Besov Space

In this paper, we give an equivalent characterization of the Besov space. This reveals the equivalent relation between the mixed derivative norm and single-variable norm. Fourier multiplier, real interpolation, and Littlewood-Paley decomposition are applied.

Interpolation theorem for Nikol’skii-Besov type spaceswith mixed metric

In this paper we study the interpolation properties of Nikol’skii-Besov spaces with a dominant mixed derivative and mixed metric with respect to anisotropic and complex interpolation methods. An interpolation theorem is proved for a weighted discrete space of vector-valued sequences l^α_q(A). It is shown that the Nikol’skii-Besov space under study is a retract of the space l^α_q(Lp). Based on the above results, interpolation theorems were obtained for Nikol’skii-Besov spaces with the dominant mixed derivative and mixed metric.

Non-local problem with integral conditions for the three-dimensional high order hyperbolic equations

В работе для трехмерного гиперболического уравнения высокого порядка с доминирующей смешанной производной исследуется разрешимость нелокальной задачи с интегральными условиями. Поставленная задача сводится к интегральному уравнению и с помощью априорных оценок доказывается существование единственного решения. In the work the solvability of the non-local problem with integral conditions is investigated for the three-dimensional high order hyperbolic equation with dominated mixed derivative. The problem is reduced to the integral equation and existence of the solution is proved by the help of aprior estimations.