An adaptive finite element method is adopted to simulate the steady state coupled Schrödinger equations with a small parameter. We use damped Newton iteration to solve the nonlinear algebraic system. When the solution domain is elliptic, our numerical results with Dirichlet or Neumann boundary conditions are consistent with previous theoretical results. For the dumbbell and circular ring domains with Dirichlet boundary conditions, we obtain some new results that may be compared with future theoretical analysis.

In this paper we prove existence and qualitative properties of solutions for a nonlinear elliptic system arising from the coupling of the nonlinear Schrödinger equation with the Poisson equation. We use a contraction map approach together with estimates of the Bessel potential used to rewrite the system in an integral form.

The Exp-function method is applied to construct a new type of solution of the coupled (2+1)-dimensional nonlinear system of Schrödinger equations. It is shown that the method provides a powerful mathematical tool for solving nonlinear evolution equations in mathematical physics.

We give the definitions of exact and approximate controllability forlinear and nonlinear Schrödinger equations, review fundamental criteria for controllability and revisit a classical “No-go” resultfor evolution equations due to Ball, Marsden and Slemrod. In Section 2 we prove corresponding results on non-controllabilityfor the linear Schrödinger equation and distributed additive control,and we show that the Hartree equation of quantum chemistry with bilinearcontrol $(E(t)\cdot x) u$ is not controllable in finite or infinite time. Finally, in Section 3, we give criteria for additive controllability of linear Schrödinger equations, andwe give a distributed additive controllability result for the nonlinear Schrödinger equation if the data are small.

In this paper we establish a variantand generalized weak linkingtheorem, which contains more delicate result and insures the existence ofboundedPalais–Smale sequences of a strongly indefinite functional.The abstract result will be used to study thesemilinear Schrödinger equation $-\Deltau+V(x)u=K(x)|u|^{2^\ast-2}u+g(x, u),u\in W^{1,2}({\bf R}^N)$ , where N ≥ 4; V,K,g are periodicin xj for 1 ≤ j ≤ N and 0 is in a gap of the spectrumof -Δ + V; K>0. If $0<g(x, u)u\leq c|u|^{2^\ast}$ for anappropriate constant c, we show that this equation has anontrivial solution.