Quantum nonlinear Schrödinger equation. II. Explicit solution
type: Item
instance of: scholarly article
title: Quantum nonlinear Schrödinger equation. II. Explicit solution
zbMATH Open document ID: 0692.35089
DOI: 10.1016/0022-1236(88)90090-0
author: Eugene Gutkin
published in: Journal of Functional Analysis
publication date: 1988-01-01T00:00:00Z
review text: [For Part I see Ann. Inst. Henri Poincaré, Anal. Nonlinéaire 3, 285- 314 (1986; Zbl 0614.35086).] This is the second in a series of papers on the nonlinear Schrödinger equation (NLS) \[ \sqrt{-1}\psi_ t=-\psi_{xx}+2c\psi^+\phi^ 2, \] an evolution equation on the time-dependent ``annihilation operators'' \(\psi\) (x,t) on the Fock spaces \(\hat {\mathcal R}=\oplus^{\infty}_{N=0}{\mathcal H}_ N.\) Explicit formulas for the solution of the NLS in terms of ``collision expansions'' are obtained. The approach is based on the method of intertwining operators for the Hamiltonian \(\hat H\) on \(\hat {\mathcal H}\) which is formally given in terms of the creation and annihilation operators \(\psi^+\), \(\psi\) by \[ \hat H=\int^{\infty}_{-\infty}dx[-\psi^+\psi_{xx}+c\psi^+\psi^ 2]. \]
Mathematics Subject Classification ID: 35Q99
Mathematics Subject Classification ID: 35K55
Mathematics Subject Classification ID: 35C05
Mathematics Subject Classification ID: 81T99
zbMATH DE Number: 4132731
zbMATH Keywords: nonlinear Schrödinger equation
zbMATH Keywords: Fock spaces
zbMATH Keywords: collision expansions
zbMATH Keywords: creation and annihilation operators
reviewed by: Joachim Weidmann
MaRDI profile type: Publication
full work available at URL: 0022-1236(88)90090-0
OpenAlex ID: W2021326193
cites work: Q5340804
cites work: Q3687118
cites work: Quantum nonlinear Schrödinger equation. I: Intertwining operators
cites work: Integrable systems with Delta-potential
cites work: Conservation laws for the nonlinear Schrödinger equation
cites work: Completely integrable systems and groups generated by reflections
cites work: Q3361250
cites work: Quantum version of the method of inverse scattering problem
cites work: Corner transfer matrices and Lorentz invariance on a lattice
cites work: Commutation relations for creation–annihilation operators associated with the quantum nonlinear Schrödinger equation
cites work: Q4066072
description: scientific article; zbMATH DE number 4132731