Project B2 • Dispersion Management (7/2015 - 6/2019)

Principal investigators

Project summary

Owing to the extraordinary increase in data transfer worldwide caused in particular by the internet, there is a great need for ultra-high-speed optical data transmission over intercontinental distances. In this field, the dispersion management technique has led to a vast increase in the data transfer rate achievable in optical cables. It is based on a strong local variation of the dispersion along the cable with a small or even zero average dispersion \(d_\mathrm{av}\). In view of its practical relevance in modern ultra-high-speed optical cables, the dispersion management technique has been extensively studied in physics with a special focus on standing waves, the so-called dispersion managed (or DM) solitons.

In a mathematically rigorous way, it has so far been possible to show basic properties such as the wellposedness for the corresponding evolution equation and (if \(d_\mathrm{av}\ge0\)) the existence of ground states, i.e., DM solitons having minimal energy. Moreover, certain aspects of the qualitative behavior of DM solitons have been established, namely their regularity and decay, as well as the stability of the set of ground states.

However, other fundamental questions about, for instance, uniqueness (modulo natural invariances), symmetry, and asymptotic stability of the ground states are still wide open. In our project we will tackle these problems, and we will also investigate more complex features of the solitons suggested by the physics literature: There is evidence that the ground states are highly oscillating in the case of small average dispersion \(d_\mathrm{av}\) and that anti-symmetric multibump solitons of higher energy should exist. We further want to show that there are no DM solitons for negative \(d_\mathrm{av}\) if \(|d_\mathrm{av}|\) is sufficiently large. So far only cubic (and very rarely quintic) nonlinearities have been studied in the literature, and we intend to also deal with more general material laws.

Publications

1. F. Bonetto, A. Geisinger, M. Loss, and T. Ried. Entropy decay for the Kac evolution. Comm. Math. Phys., 363(3):847–875, November 2018. URL https://doi.org/10.1007/s00220-018-3263-0. [preprint] [bibtex]

2. D. Hundertmark, Y.-R. Lee, T. Ried, and V. Zharnitsky. Solitary waves in nonlocal NLS with dispersion averaged saturated nonlinearities. J. Differential Equations, 265(8):3311–3338, October 2018. URL https://doi.org/10.1016/j.jde.2017.08.028. [preprint] [bibtex]

3. J.-M. Barbaroux, D. Hundertmark, T. Ried, and S. Vugalter. Strong smoothing for the non-cutoff homogeneous Boltzmann equation for Maxwellian molecules with Debye–Yukawa type interaction. Kinet. Relat. Models, 10(4):901–924, December 2017. URL https://doi.org/10.3934/krm.2017036. [preprint] [bibtex]

4. M.-R. Choi, D. Hundertmark, and Y.-R. Lee. Discrete diffraction managed solitons: threshold phenomena and rapid decay for general nonlinearities. J. Math. Phys., 58(10):101513, 43, October 2017. URL https://doi.org/10.1063/1.5004253. [preprint] [bibtex]

5. D. Hundertmark, Y.-R. Lee, T. Ried, and V. Zharnitsky. Dispersion managed solitons in the presence of saturated nonlinearity. Phys. D, 356-357:65–69, October 2017. URL https://doi.org/10.1016/j.physd.2017.06.004. [preprint] [bibtex]

6. J.-M. Barbaroux, D. Hundertmark, T. Ried, and S. Vugalter. Gevrey smoothing for weak solutions of the fully nonlinear homogeneous Boltzmann and Kac equations without cutoff for Maxwellian molecules. Arch. Ration. Mech. Anal., 225(2):601–661, August 2017. URL https://doi.org/10.1007/s00205-017-1101-8. [preprint] [bibtex]

7. M.-R. Choi, D. Hundertmark, and Y.-R. Lee. Thresholds for existence of dispersion management solitons for general nonlinearities. SIAM J. Math. Anal., 49(2):1519–1569, April 2017. URL https://doi.org/10.1137/15M103666X. [bibtex]

Theses

8. T. Ried. On some nonlinear and nonlocal effective equations in kinetic theory and nonlinear optics. PhD thesis, Karlsruhe Institute of Technology (KIT), October 2017. [bibtex]