A coupled nonlinear Schrödinger–Poisson equation is considered which contains a time‐dependent dissipation function as a specific model of dissipation effects in nonlinear quantum transport theory and other areas. The Wigner–Poisson equation associated with this system is derived. Using conservation and quasiconservation laws and certain growth assumptions for the nonlinearities and the dissipation function, global existence of solutions to the Cauchy problem of the time‐dependent Schrödinger–Poisson system is shown both for small (attractive case) or arbitrary data (repulsive case).
REFERENCES
1.
N.
Akhmediev
, A.
Ankiewicz
, and J. M.
Soto-Crespo
, Opt. Lett.
18
, 411
(1993
);2.
F.
Brezzi
and P. A.
Markowich
, Math. Methods Appl. Sci.
14
, 35
(1991
).3.
R.
Illner
, H.
Lange
, and P. F.
Zweifel
, Math. Methods Appl. Sci.
17
, 349
(1994
).4.
5.
P. Markowich, C. Ringhofer, and C. Schmeiser, Semiconductor Equations (Springer-Verlag, New York, 1990).
6.
H.
Lange
and P. F.
Zweifel
, “Dissipation in Wigner-Poisson systems
,” J. Math Phys.
35
, 1513
(1994
).7.
8.
9.
A. Arnold and C. Ringhofer, “An operator splitting method for the Wigner-Poisson problem,” Preprint 174 Center for Appl. Math., Purdue Univ., 1992.
10.
11.
“T. Cazenave, An Introduction to Nonlinear Schrödinger Equations, Textos de Métodos Matemáticos 22 (Univ. Fed. do Rio de Janeiro, CCMN, Instituto de Matemática, 1989).
12.
A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations (Springer-Verlag, Berlin, 1983).
13.
A. Friedman, Partial Differential Equations (Rinehart and Winston, New York, 1969).
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© 1995 American Institute of Physics.
1995
American Institute of Physics
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