Semiclassical limit and well-posedness of nonlinear Schrödinger-Poisson systems. (English) Zbl 1055.35111

Li, Hailiang; Lin, Chi-Kun

Electron. J. Differ. Equ. 2003, Paper No. 93, 17 p. (2003).

The authors deal with the Cauchy problem for the nonlinear Schrödinger-Poisson system \[ \begin{gathered} i\varepsilon\psi^\varepsilon_t+ {\varepsilon^2\over 2} \Delta\psi^\varepsilon- (V^\varepsilon(x,t)+ f'(|\psi^\varepsilon|^2) \psi^\varepsilon- (\arg\psi^\varepsilon) \psi^\varepsilon= 0,\\ -\Delta V^\varepsilon= |\psi^\varepsilon|^2- C(x),\quad V\to 0\text{ as }| x|\to\infty,\end{gathered}\tag{1} \] subject to the rapidly oscillating (WKB) initial condition \[ \psi^\varepsilon(x,0)= \psi^\varepsilon_0(x)= A^\varepsilon_0(x) \exp\biggl({i\over\varepsilon} S_0(x)\biggr),\tag{2} \] where \(f\), \(S_0\), \(A^\varepsilon_0\) are given functions. Under some suitable assumptions on the data (1)–(2) the authors show the local well-posedness. Moreover, they obtain global existence and study the time-asymptotic behaviour of classical solutions of (1)–(2).

Reviewer: Messoud A. Efendiev (Berlin)

MSC:

35Q55	NLS equations (nonlinear Schrödinger equations)
81Q20	Semiclassical techniques, including WKB and Maslov methods applied to problems in quantum theory
35A05	General existence and uniqueness theorems (PDE) (MSC2000)
76Y05	Quantum hydrodynamics and relativistic hydrodynamics

Keywords:

Schrödinger-Poisson system; quantum hydrodynamics; Euler-Poisson system; semiclassical limit; WKB expansion; quasilinear symmetric hyperbolic system; Cauchy problem