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

We prove the existence of periodic solutions and solitons in the nonlinear Schrödinger 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.

► A nonlinear Schrödinger equation with a nonlocal integral term is considered. ► Periodic solutions and solitary waves under mild assumptions. ► The problem is written as a fixed point problem on a suitable Banach space. ► The proof combines a fixed point theorem with properties of the Green’s function.