The Solvability of the Initial-Boundary
Value Problems for a Nonlinear Schrödinger Equation with a Special Gradient Term

G. Yagub

Kafkas University, Paşaçayırı Campus, Kars, 36040, Turkey
E-mail: gabilya@mail.ru

N.S. Ibrahimov

Baku State University, 23 Academic Zahid Khalilov St., Baku, AZ 1148, Azerbaijan;
Lankaran State University, 50 Hazi Aslanov St., Lankaran, AZ 4200, Azerbaijan
E-mail: natiq ibrahimov@mail.ru

Received February 13, 2017, revised January 10, 2018.

Abstract

In this paper, the initial-boundary value problems for the two-dimensional nonlinear Schrödinger equation with a special gradient term with purely imaginary coefficients in the nonlinear part, when the coefficients of the equation are measurable bounded functions, are considered. The existence
and uniqueness of solutions of the first and second initial-boundary value problems is proved almost everywhere.

Mathematics Subject Classification 2010: 35D, 35M, 35Q. Key words: Schrödinger equation, special gradient term, existence and
uniqueness, first and second initial-boundary value problems.