An integrable time-dependent non-linear Schrödinger equation

نویسنده

  • P. A. HORVÁTHY
چکیده

The cubic non-linear Schrödinger equation (NLS), where the coefficient of the non-linear term can be a function F (t, x), is shown to pass the Painlevé test of Weiss, Tabor, and Carnevale only for F = (a+ bt), where a and b constants. This is explained by transforming the time-dependent system into the ordinary NLS (with F = const.) by means of a time-dependent non-linear transformation, related to the conformal properties of non-relativistic space-time. (7/2/2008) Physics Letter A (submitted). () e-mail: [email protected] () e-mail: [email protected] 2 Horváthy & Yera Let us consider the cubic non-linear Schrödinger equation (NLS), (1) iut + uxx + F (t, x)|u|u = 0, where u = u(t, x) is a complex function in 1+1 space-time dimension. When the coefficient F (t, x) of the non-linearity is a constant, this is the usual NLS, which is known to be integrable. But what happens, when the coefficient F (t, x) is a function rather then just a constant ? Performing the Painlevé analysis of Weiss, Tabor and Carnevale [1], we show Theorem1 : The generalized non-linear Schrödinger equation (1) only passes the Painlevé test if the coefficient of the non-linear term is of the form (2) F (t, x) = 1 at+ b , a, b = const. Proof. As it is usual in studying non-linear Schrödiger-type equations [2], we consider Eqn. (1) together with its complex conjugate (v = u), (3) iut + uxx + Fu v = 0, −ivt + vxx + Fvu = 0. This system will pass the Painlevé test if u et v have generalised Laurent series expansions, (4) u = +∞ ∑

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تاریخ انتشار 1998