Moving Gap Solitons in Deep Periodic Structures with Transversally Crossing Bands

• Applicant: Professor Tomas Dohnal, Ph.D.
• Subject Area: Mathematics
• Term: from 2014 to 2017
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 253403347

• Final Report Year: 2017

Final Report Abstract

Firstly, one dimensional nonlinear wave problems in periodic media have been considered. Wave-packets coupling two Bloch waves with opposite group velocities and equal frequencies have been shown to be approximated by the coupled mode equations (CMEs) of Dirac type. This holds, contrary to the original expectation, even if the two Bloch waves do not lie at a Dirac point (transverse crossing point) of the band structure. The CME approximation was rigorously justified for the case of the one dimensional nonlinear Schrödinger equation with a periodic potential (PNLS) as the original system. The CMEs were shown numerically to possess a family of moving solitary waves parameterized by the velocity. These generate approximate solitary waves of PNLS. For the one dimensional wave equation we considered the question of the existence of exact localized breathers using the tools of spatial dynamics. This problem, however, remains open. In higher dimensional problems we have considered first the case of wave-packets with a single carried Bloch wave. For the case of the n-dimensional PNLS we proved the approximation by a constant coefficient nonlinear Schrödinger equation. Examples of approximate solitary waves of the PNLS have been numerically computed. Second, in another asymptotic setting, wave-packets coupling Bloch waves with different group velocities have been shown to be approximated by n-dimensional CMEs. However, contrary to the one dimensional case, these do not seem to possess traveling solitary waves. Nevertheless, standing solitary waves have been found for certain CMEs. Also here the justification has been carried out. A numerical treatment of waves on unbounded domains requires suitable (radiation) boundary conditions on the computational domain. For the time harmonic case, i.e. the Helmholtz equation, in a two dimensional wave-guide and a periodic medium we have designed a novel radiation boundary condition by expanding the solution near the boundary in terms of Bloch waves with outgoing group velocities.

Publications

• Longtime Behaviour of Nonlinear Waves, Bielefeld, 8.-12.6.2015. “Justification of Coupled Mode Equations for Wavepackets in the Periodic NLS with Finite Contrast”
  L. Helfmeier
  
• “Movement of Gap Solitons across Deep Gratings in the Periodic NLS,” Longtime Behaviour of Nonlinear Waves, Bielefeld, 8.-12.6.2015
  T. Dohnal
  
• 4th Workshop of the GAMM Activity Group on Analysis of Partial Differential Equations, Dortmund, 25.-28.9.2016. “Coupled Mode Asymptotics for Wavepackets in the Periodic Nonlinear Schrödinger Equation”
  T. Dohnal
  
• Mathematical and Computational Aspects of Maxwell’s Equations, Durham, 11.-21.7.2016. “Dirac Type Asymptotics for Wavepackets in Periodic Media of Finite Contrast”
  T. Dohna
  
• A Bloch wave numerical scheme for scattering problems in periodic wave-guides, submitted, 2017 (25 pages)
  T. Dohnal and B. Schweizer
  
• International Conference on Elliptic and Parabolic Problems, Gaeta, 22-26.5.2017. “Justification of the Coupled mode asymptotics for localized wavepackets in PNLS”
  Helfmeier, L.
  
• Justification of the Coupled Mode Asymptotics for Localized Wavepackets in the Periodic Nonlinear Schrödinger Equation, J. Math. Anal. Appl., 450:691-726 (2017)
  T. Dohnal and L. Helfmeier
  (See online at https://doi.org/10.1016/j.jmaa.2017.01.039)
• NLS approximation for wavepackets in periodic cubically nonlinear wave problems in Rd (35 pages)
  T. Dohnal and D. Rudolf