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Numerical aspects of nonlinear Schrödinger equations in the presence of caustics. (English) Zbl 1162.35068

The authors introduce a numerical study of the semiclassical solutions to the following nonlinear Schrödinger equation with \(\varepsilon \ll 1:\) \[ \begin{aligned} i\varepsilon \partial_{t} u^{\varepsilon} + \frac{\varepsilon^{2}}{2}\Delta u^{\varepsilon} & = |u^{\varepsilon}|^{2\sigma}u^{\varepsilon}, \quad (t, x)\in \mathbb{R}_{+}\times \mathbb{R}^{n},\\ u^{\varepsilon}|_{t=0} & = \varepsilon^{p} f(x) e^{i\phi_{0}(x)/\varepsilon} \end{aligned} \] when a caustic (a focal point or a cusp) is formed (i.e. beyond breakup time).
First, rigorous results concerning the semiclassical limit of the free Schrödinger equation are reviewed, the scattering operator theory is recalled. Second, numerical experimentrs are carried out on the focal point singularity for which several results have been proved rigorously. Furthermore, the scattering operator is numerically studied. Finally, experiments on the cusp caustic are displayed, and similarities with the focal point are discussed. Several shortcoming of spectral time-splitting schemes are investigated.

MSC:

35Q55 NLS equations (nonlinear Schrödinger equations)
65Z05 Applications to the sciences
65M99 Numerical methods for partial differential equations, initial value and time-dependent initial-boundary value problems
81Q20 Semiclassical techniques, including WKB and Maslov methods applied to problems in quantum theory
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