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Geometric optics with caustic crossing for some nonlinear Schrödinger equations. (English) Zbl 0970.35143

The initial value problem \[ i\varepsilon\partial u^\varepsilon/\partial t+ \textstyle{{1\over 2}} \varepsilon^2\Delta u^\varepsilon= \lambda\varepsilon^\alpha|u^\varepsilon|^\beta u^\varepsilon, \]
\[ u^\varepsilon(x, 0)= e^{-ix^2/2\varepsilon} f(x) \] is thoroughly investigated, where \(\varepsilon\) is a parameter going to zero, \(f\) is smooth, \(\lambda\) is real, \(\alpha\geq 1\) and \(\beta> 0\). The problem is concerned with linear (nonlinear) propagation if \(\alpha> 1\) \((\alpha=1)\) and with linear (nonlinear) caustics if \(\alpha> n\beta/2\) \((\alpha= n\beta/2)\). The mentioned four possibilities are separately dicussed by using the WKB method and generalized Laplace integrals \[ u^\varepsilon(x, t)= \varepsilon^{-n/2} \int e^{-i(t-1)\xi^2/2\varepsilon+ ix\xi/\varepsilon+ ig(t,\xi)} a_\varepsilon(t, \xi) d\xi \] (\(g=0\) in the linear case). The resulting caustic crossing is affected by the boundary layer (for the nonlinear case) in which the crossing is described by a scattering operator. (Many technicalities prevent to state more results of this remarkable article).

MSC:

35Q55 NLS equations (nonlinear Schrödinger equations)
78A05 Geometric optics
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