NLS approximation for wavepackets in periodic cubically nonlinear wave problems in ℝ^d

ABSTRACT

The dynamics of single carrier wavepackets in nonlinear wave problems over periodic structures can be often formally approximated by the constant coefficient nonlinear Schrödinger equation as an effective model for the wavepacket envelope. We provide a detailed proof of this approximation result for the Gross–Pitaevskii (GP) equation and a semilinear wave equation, both with periodic coefficients in $\mathbb{N}\ni d$ spatial dimensions and with cubic nonlinearities. The proof is carried out in Bloch expansion variables with estimates in an $L^1$-type norm, which translates to an estimate of the supremum norm of the error. The regularity required from the periodic coefficients in order to ensure a small residual and a small error is discussed. We also present a numerical example in two spatial dimensions confirming the approximation result and presenting an approximate traveling solitary wave in the GP with periodic coefficients.

KEYWORDS:

• Wavepackets
• NLS approximation
• Bloch waves
• periodic media
• solitary wave
• approximation error
• Gross–Pitaevskii
• nonlinear wave equation

MSC:

• 35Q55
• 35Q60
• 35L71
• 41A60

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

This research is supported by the German Research Foundation, DFG-Deutsche Forschungsgemeinschaft [grant number DO1467/3-1]. D. R. is supported by the SFB/TRR 191 ‘Symplectic Structures in Geometry, Algebra and Dynamics’, funded by the DFG.