Existence of periodic and solitary waves for a nonlinear Schrِdinger equation with nonlocal integral term of convolution type

We prove the existence of periodic solutions and solitons in the nonlinear Schrِdinger equation with a nonlocal integral term of convolution type. By separating phase and amplitude, the problem is reduced to an integro-differential formulation that can be written as a fixed point problem for a suitable operator on a Banach space. Then a fixed point theorem due to Krasnoselskii can be applied.