Equivalence and nonexistence of standing waves for coupled Schrödinger equations with Chern-Simons gauge fields

Abstract

This paper is devoted to the study of standing waves for so-called $N=2$ supersymmetric Chern-Simons-Schrödinger equations, a coupled system of Schrödinger equations in which Chern-Simons gauge fields are incorporated.We show that there is no nontrivial standing wave to the $N=2$ supersymmetric Chern-Simons-Schrödinger equations when the coupling constants are less than critical numbers.We also prove that static $N=2$ supersymmetric Chern-Simons-Schrödinger equations are equivalent to their first order self-dual system when the coupling constants are critical.