Dynamics of the solution for Chern–Simons–Schrödinger system under Coulomb gauge condition

In this paper, we analyze new dynamical phenomena of the solutions to the Cauchy problem of Schrödinger equations with Chern–Simons gauge field. This model arises in the non-relativistic gauge theory and captures a strong nonlocal effect due to the presence of the covariant derivative operators. We study the existence and classification of the initial data in $$H^1({\mathbb {R}}^2)$$ H 1 ( R 2 ) which generate either global or finite time blow-up solution to the problem under the Coulomb gauge condition, by developing a delicate variational characterization and some suitable scaling properties of the energy functional. In particular, we prove that there exist $$\phi \in H^1({\mathbb {R}}^2)$$ ϕ ∈ H 1 ( R 2 ) and real numbers $$\tau _1 \tau _2$$ τ > τ 2 . The Chern–Simons gauge potentials make the problem interesting and different from the classical Schrödinger equations.