<p>Periodic waves, kink and anti-kink wave solutions in a nonlinear Schrödinger equation with distributed delay are investigated in this paper. The traveling wave solutions are considered in two cases. We provided conditions not only for the uniqueness but also for the existence of two periodic wave solutions. A near-Hamiltonian system is formed by reducing the singularly perturbed system on a locally invariant manifold. When all coefficients of the nonlinear Landau term are non-vanishing, the Melnikov function comprises three Abelian integrals. By analyzing the properties of a linear combination of two ratio functions involving these integrals, we prove that the Melnikov function possesses two simple zeros. This result implies the persistence of two limit cycles and, consequently, two periodic wave solutions. When linear coefficient of the Landau term is zero, the Melnikov function has two Abelian integrals, we prove the monotonicity and range of ratio function with two Abelian integrals, which leads to one limit cycle, and then derives one periodic wave solution. Further, the persistence of kink (anti-kink) wave solution is proved by heteroclinic bifurcation theory. We exhibit numerical simulations to illustrate the given results.</p>