We study standing waves for nonlinear Schrödinger equations with the gauge field. Some existence results of standing waves are established by applying variational methods to the functional which is obtained by representing the gauge field A μ in terms of complex scalar field φ. We also show that there exists no standing wave for certain range of parameters by establishing a new inequality of Sobolev type.

We prove the existence of infinitely many radially symmetric standing-wave solutions of the Chern-Simons-Schrödinger system. Our result is established by applying the mountain-pass theorem to the functional, which is obtained by representing gauge fields Aμ in terms of a scalar field ϕ.

We study finite time blow-up solutions of the Chern-Simons-Schrödinger system. In particular, explicit blow-up solutions are constructed by observing the pseudo-conformal invariance and finding solutions of the self-dual equations.

Abstract. In this paper, we address the problem of local well-posedness of the Chern-SimonsDirac (CSD) and the Chern-Simons-Higgs (CSH) equations in the Lorenz gauge for low regularity initial data. One of our main contributions is the uncovering of a null structure of (CSD). Combined with the standard machinery of X s,b spaces, we obtain local well-posedness of (CSD) for initial data aµ, ψ ∈ H 1/4+ǫ x . Moreover, it is observed that the same techniques applied to (CSH) lead to a quick proof of local-wellposedness for initial data aµ ∈ H

We prove that the Chern-Simons-Schrödinger system, under the condition of a Coulomb gauge, has a unique local-in-time solution in the energy space 1 (R 2). The Coulomb gauge provides elliptic features for gauge fields 0 ,. The Koch-and Tzvetkov-type Strichartz estimate is applied with Hardy-Littlewood-Sobolev and Wente's inequalities.