Conservation laws for the nonlinear Schrödinger equation. (English) Zbl 0585.35080

Gutkin, Eugène

Ann. Inst. Henri Poincaré, Anal. Non Linéaire 2, 67-74 (1985).

The purpose of the paper is to obtain a method which allows to derive conservation laws of the quantum nonlinear Schrödinger equation \(i \psi_ t=-\psi_{xx}+2c \psi^{†}\psi^ 2\) where \(\psi\) is interpreted as a two dimensional quantum field. The heart of the method is the construction of an operator P which intertwines the Laplacian with boundary conditions corresponding to the Hamiltonian \((\partial /\partial x_{k+1}-\partial /\partial x_ k)F=cF,\) \(c>0\) with the Laplacian with Neumann boundary conditions \((\partial /\partial x_{k+1}-\partial /\partial x_ k)F=0.\)
Of the infinitely many conservation laws the first four are derived explicitly. The fourth one shows a difference to the analogous classical one. The three first conservation laws reproduce an earlier result by H. B. Thacker [Phys. Rev. D 17, 1031 (1978)].

Reviewer: H.Siedentop

MSC:

35Q99	Partial differential equations of mathematical physics and other areas of application
81T08	Constructive quantum field theory
35G20	Nonlinear higher-order PDEs

Keywords:

conservation laws; quantum nonlinear Schrödinger equation

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