Justification of the Nonlinear Schrödinger approximation of nonlinear dispersive systems
Dienstag, 04.02.2020, 10:30 Uhr
Wolf-Patrick Düll (Uni Stuttgart)
In many physical systems dispersion plays an important role. If the systems are nonlinear, it is possible that concentration effects balance dispersive effects such that structures of permanent form like solitons can be observed.
A famous nonlinear PDE which models this phenomenon is the Nonlinear Schrödinger (NLS) equation. Moreover, the NLS equation can be formally derived as an approximation equation for the dynamics of the envelopes of oscillating wave packets in complicated nonlinear dispersive systems. The so-called NLS approximation has various applications in science and technology, for example, in hydrodynamics, optics or spintronics. To understand to which extent this approximation yield correct predictions of the qualitative behavior of the original systems it is important to justify the validity of the approximation by estimates of the approximation errors in the physically relevant length and time scales.
In this talk, we give an overview on the NLS approximation, its applications and its justifications. Concerning the justifications, we will put special emphasis on the most challenging case, namely, the proof of error estimates for the NLS approximation being valid for surface water waves with and without surface tension. These estimates are obtained by parametrizing the two-dimensional surface waves by arc length, which enables us to derive error bounds that are uniform with respect to the strength of the surface tension, as the height of the wave packet and the surface tension go to zero.
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