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Localized States and Dynamics in the Nonlinear Schrödinger/Gross-Pitaevskii Equation

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Dynamics of Partial Differential Equations

Part of the book series: Frontiers in Applied Dynamical Systems: Reviews and Tutorials ((FIADS,volume 3))

Abstract

Nonlinear dispersive waves are wave phenomena resulting from the interacting effects of nonlinearity and dispersion. Dispersion refers to the property that waves of different wavelengths travel at different velocities. This property may, for example, be due to the material properties of the medium, e.g. chromatic dispersion [1], or to the geometric arrangement of material constituents, e.g. a periodic medium with Floquet-Bloch band dispersion [3]. Nonlinearity distorts the shape of a localized structure or creates amplitude dependent non-uniformities in phase. This may concentrate or localize energy in a region of space (attractive/focusing nonlinearity) or tend to expel energy from compact sets (repulsive/defocusing nonlinearity). In electromagnetics, nonlinearities may arise due to the intensity dependence of dielectric parameters (e.g., Kerr effect [1]). Physical phenomena in which the effects of both dispersion and nonlinearity play a role are ubiquitous. Some examples are: (a) long waves of small amplitude at a water–air interface [114], (b) a nearly mono-chromatic laser beam propagating through air, glass, or water [36], (c) light-pulses propagating through optical fiber waveguides [1], and (d) the macroscopic dynamics of weakly correlated quantum particles in a Bose-Einstein condensate; see, for example, [30, 49, 88]. Interest in nonlinear dispersive waves and their interaction with nonhomogeneous media ranges from Fundamental to Applied Science with great promise in engineering/technological applications due to major advances in materials science and micro- and nano-structure fabrication techniques; see, for example, [109].

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Notes

  1. 1.

    This work is supported in part by U.S. NSF grant DMS-10-08855 and DMS-14-12560.

  2. 2.

    \(\Gamma _{0}(z,z^{{\ast}})\) assumed to be strictly positive, is non-negative by the following:

    $$\displaystyle\begin{array}{rcl} & & \mathfrak{I}\ [-\Delta + V -\omega _{\star } - i0]^{-1} = \frac{1} {2i}\ \lim _{\delta \downarrow 0}\ \left (\ [-\Delta + V -\omega _{\star } - i\delta ]^{-1} - [-\Delta + V -\omega _{\star } + i\delta ]^{-1}\ \right ), {}\\ & & =\ \ \pi \ \delta \left (-\Delta + V -\omega _{\star }\right ),\ \ \mathrm{where}\ \omega _{\star } \in \sigma _{cont}(-\Delta + V ). {}\\ \end{array}$$

    Note: \(\mathfrak{I}\ [-\Delta + V -\omega _{\star } - i0]^{-1}\) projects onto the generalized mode at energy \(\omega _{\star } \in \sigma _{cont}(-\Delta + V ).\)

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Acknowledgements

The author would like to thank J. Marzuola and E. Shlizerman for stimulating discussions and helpful comments on this chapter. He also wishes to thank the referees for their careful reading and suggested improvements. This work was supported, in part, by NSF Grants DMS-1008855 and DMS-1412560, and a grant from the Simons Foundation (#376319).

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    Michael I. Weinstein

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Weinstein, M.I. (2015). Localized States and Dynamics in the Nonlinear Schrödinger/Gross-Pitaevskii Equation. In: Dynamics of Partial Differential Equations. Frontiers in Applied Dynamical Systems: Reviews and Tutorials, vol 3. Springer, Cham. https://doi.org/10.1007/978-3-319-19935-1_2

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