On the inverse spectral problem for the quasi-periodic Schrödinger equation

We study the quasi-periodic Schrödinger equation $$-\psi''(x) + V(x) \psi(x) = E \psi(x), \quad x \in{ \mathbf {R}} $$ in the regime of “small” V. Let $(E_{m}',E''_{m})$ , m∈Z ^ν , be the standard labeled gaps in the spectrum. Our main result says that if $E''_{m} - E'_{m} \le\varepsilon\exp(-\kappa_{0} |m|)$ for all m∈Z ^ν , with ε being small enough, depending on κ ₀>0 and the frequency vector involved, then the Fourier coefficients of V obey $|c(m)| \le \varepsilon^{1/2} \exp(-\frac{\kappa_{0}}{2} |m|)$ for all m∈Z ^ν . On the other hand we prove that if |c(m)|≤εexp(?κ ₀|m|) with ε being small enough, depending on κ ₀>0 and the frequency vector involved, then $E''_{m} - E'_{m} \le2 \varepsilon\exp(-\frac {\kappa_{0}}{2} |m|)$ .     

Transfer of energy to high frequencies in the cubic defocusing nonlinear Schrödinger equation

We consider the cubic defocusing nonlinear Schrödinger equation on the two dimensional torus. We exhibit smooth solutions for which the support of the conserved energy moves to higher Fourier modes. This behavior is quantified by the growth of higher Sobolev norms: given any δ?1,K?1,s>1, we construct smooth initial data u ₀ with \(\|u_{0}\|_{{H}^{s}}<\delta\), so that the corresponding time evolution u satisfies \(\|u(T)\|_{{H}^{s}}>K\) at some time T. This growth occurs despite the Hamiltonian’s bound on \(\|u(t)\|_{\dot{H}^{1}}\) and despite the conservation of the quantity \(\|u(t)\|_{L^{2}}\).The proof contains two arguments which may be of interest beyond the particular result described above. The first is a construction of the solution’s frequency support that simplifies the system of ODE’s describing each Fourier mode’s evolution. The second is a construction of solutions to these simpler systems of ODE’s which begin near one invariant manifold and ricochet from arbitrarily small neighborhoods of an arbitrarily large number of other invariant manifolds. The techniques used here are related to but are distinct from those traditionally used to prove Arnold Diffusion in perturbations of Hamiltonian systems.     

Fractional Schrödinger equation; solvability and connection with classical Schrödinger equation

We consider the Dirichlet boundary problem for semilinear fractional Schrödinger equation with subcritical nonlinear term. Local and global in time solvability and regularity properties of solutions are discussed. But our main task is to describe the connections of the fractional equation with the classical nonlinear Schrödinger equation, including convergence of the linear semigroups and continuity of the nonlinear semigroups when the fractional exponent α approaches 1.     

Dirichlet problem for Schrödinger equation with the boundary value in the BMO space

Science China Mathematics - Let (X, d, μ) be a metric measure space satisfying a Q-doubling condition (Q &gt; 1) and an L2-Poincaré inequality. Let $${\cal L} = {\cal L} + V$$ be a...     

Butterfly-like waves in the nonlinear Schrödinger equation with a combined dispersion term

All possible optical waves in the nonlinear Schrödinger equation with a combined dispersion term are determined according to different parameters’ regions. First, we find this equation has some popular exotic solutions, such as peaked waves, looped and cusped waves. What is more interesting, this equation admits some very particular waves, such as double kinked waves and butterfly-like waves. These new two types of solutions have not been reported in the literature regarding the study of other nonlinear equations.     

Eigenfunction expansions for the Schrödinger equation with an inverse-square potential

We consider the one-dimensional Schrödinger equation -f″ + q_κf = Ef on the positive half-axis with the potential q_κ(r) = (κ² - 1/4)r^-2. For each complex number ν, we construct a solution u_ν^κ(E) of this equation that is analytic in κ in a complex neighborhood of the interval (-1, 1) and, in particular, at the “singular” point κ = 0. For -1 < κ < 1 and real ν, the solutions u_ν^κ(E) determine a unitary eigenfunction expansion operator U_κ,ν: L₂(0,∞) → L₂(R, Vκ,ν), where Vκ,ν is a positive measure on R. We show that every self-adjoint realization of the formal differential expression -?_r² + q_κ(r) for the Hamiltonian is diagonalized by the operator U_κ,ν for some ν ∈ R. Using suitable singular Titchmarsh–Weyl m-functions, we explicitly find the measures V_κ,ν and prove their continuity in κ and ν.     

Well-posedness of the two-dimensional nonlinear Schrödinger equation with concentrated nonlinearity

We consider a two-dimensional nonlinear Schrödinger equation with concentrated nonlinearity. In both the focusing and defocusing case we prove local well-posedness, i.e., existence and uniqueness of the solution for short times, as well as energy and mass conservation. In addition, we prove that this implies global existence in the defocusing case, irrespective of the power of the nonlinearity, while in the focusing case blowing-up solutions may arise.     

Soliton dynamics for the nonlinear Schrödinger equation with magnetic field

The semiclassical regime of a nonlinear focusing Schrödinger equation in presence of non-constant electric and magnetic potentials V, A is studied by taking as initial datum the ground state solution of an associated autonomous stationary equation. The concentration curve of the solutions is a parameterization of the solutions of the second order ordinary equation \({\ddot x=-\nabla V(x)-\dot x\times B(x)}\), where \({B=\nabla\times A}\) is the magnetic field of a given magnetic potential A.     

Local well-posedness for the Maxwell-Schrödinger equation

Time local well-posedness for the Maxwell-Schrödinger equation in the Coulomb gauge is studied in Sobolev spaces by the contraction mapping principle. The Lorentz gauge and the temporal gauge cases are also treated by the gauge transform.     

Space-time scattering for the Schrödinger equation

Results are obtained on the scattering theory for the Schrödinger equation $i\partial _t u(t,x) = - \Delta _x u(t,x) + V(t,x)u(t,x) + F(u(t,x))$ in spacesL ^r (R;L ^q (R ^d )) for a certain range ofr, q, the so-called space-time scattering. In the linear case (i.e.F≡)) the relation with usual configuration space scattering is established.     

Normal form for the symmetry-breaking bifurcation in the nonlinear Schrödinger equation

We derive and justify a normal form reduction of the nonlinear Schrödinger equation for a general pitchfork bifurcation of the symmetric bound state that occurs in a double-well symmetric potential. We prove persistence of normal form dynamics for both supercritical and subcritical pitchfork bifurcations in the time-dependent solutions of the nonlinear Schrödinger equation over long but finite time intervals.     

Nonlinear Schrödinger equation for the twisted Laplacian in the critical case

In [6] and [7], we prove well-posedness of solution to the nonlinear Schrödinger equation associated to the twisted Laplacian on ? ⁿ for a general class of nonlinearities including power type with subcritical case 0 ≤ α < 2/n?1. In this paper, we consider the critical case α = 2/n?1 with n ≥ 2. Our approach is based on truncation of the given nonlinearity G, which is used in [3]. We obtain solution for the truncated problem. We obtain solution to the original problem by passing to the limit.