Differential and Integral Equations

Derivation of a nonlinear Schrödinger equation with a general power-type nonlinearity in $d=1,2$

Zhihui Xie

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Abstract

In this paper, we study the derivation of a certain type of NLS from many-body interactions of bosonic particles in $d=1,2$. We consider a model with a finite linear combination of $n$-body interactions and obtain that the $k$-particle marginal density of the BBGKY hierarchy converges when particle number goes to infinity. Moreover, the limit solves a corresponding infinite Gross-Pitaevskii hierarchy. We prove the uniqueness of factorized solution to the Gross-Pitaevskii hierarchy based on a priori space time estimates. The convergence is established by adapting the arguments originated or developed in [6], [15] and [2]. For the uniqueness part, we expand the procedure in [16] by introducing a different board game argument to handle the factorial in the number of terms from Duhamel expansion. The space time bound assumption in [16] is removed in our proof.

Article information

Source
Differential Integral Equations, Volume 28, Number 5/6 (2015), 455-504.

Dates
First available in Project Euclid: 30 March 2015

Permanent link to this document
https://projecteuclid.org/euclid.die/1427744097

Mathematical Reviews number (MathSciNet)
MR3328130

Zentralblatt MATH identifier
1363.35358

Subjects
Primary: 35Q55: NLS-like equations (nonlinear Schrödinger) [See also 37K10] 81V70: Many-body theory; quantum Hall effect

Citation

Xie, Zhihui. Derivation of a nonlinear Schrödinger equation with a general power-type nonlinearity in $d=1,2$. Differential Integral Equations 28 (2015), no. 5/6, 455--504. https://projecteuclid.org/euclid.die/1427744097


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