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Quantum nonlinear Schrödinger equation. II. Explicit solution. (English) Zbl 0692.35089
[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 \[ \hat H=\int^{\infty}_{-\infty}dx[-\psi^+\psi_{xx}+c\psi^+\psi^ 2]. \]
Reviewer: J.Weidmann

MSC:
35Q99 Partial differential equations of mathematical physics and other areas of application
35K55 Nonlinear parabolic equations
35C05 Solutions to PDEs in closed form
81T99 Quantum field theory; related classical field theories
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