Nonlocal Variational Problems in Nonlinear Optics

Dispersion Management Solitons

The dispersion managed nonlinear Schrödinger equation (DM NLS) by now is a well established model in nonlinear science. Initially, the main motivation to study this equation came from fibre optics applications, after the introduction of the dispersion compensation technique (which itself appeared due to the invention of fibres with anomalous dispersion). Nowadays, DM NLS became a paradigm of a nonlinear dispersive equation with periodically varying coefficients that in some regime, e.g. strong dispersion management, leads to a dispersion averaged nonlinearity. The dispersion management equation arises from a non-autonomous 1d nonlinear Schrödinger equation and poses interesting mathematical challenges due to the nonlocal character of the nonlinearity. It contains only strongly oscillating terms, and no a priori decay is present. The interesting case of vanishing average dispersion is a singular limit. Breather-type dispersion managed solitons can be found as minimisers of a suitable non-local variational problem, whose Euler-Lagrange equation is the dispersion management equation. This variational problem, however, is invariant under a large non-compact group of transformations, which leads to a loss of compactness, since minimising sequences can easily converge weakly to zero. By a rather direct approach, using tightness properties of minimising sequences modulo the natural symmetries, existence of minimisers has been shown by Hundertmark and Lee for zero average dispersion and all physically relevant dispersion profiles. This technique was recently extended to more general non-linearities (limited essentially by the applicability of the Strichartz inequality and an Ambrosetti-Rabinowitz-type condition) to prove the existence of minimisers. We were able to extend the existence results to include saturating non-linearities, for which the Ambrosetti-Rabinowitz condition does not hold globally. Based upon a version of Ekeland's variational principle, we construct modified minimising sequences, for which compactness can be recovered by strict sub-additivity properties of the ground state energy. Using Nehari-manifold techniques these existence results can be extended to an even more general class of nonlinearities.

Related publications

  1. Solitary waves in nonlocal NLS with dispersion averaged saturated nonlinearities. Journal of Differential Equations 265 (2018), 3311-3338.
    (with D. Hundertmark, Y.-R. Lee, V. Zharnitsky)
  2. Dispersion managed solitons in the presence of saturated nonlinearity. Physica D: Nonlinear Phenomena 356–357 (2017), 65-69.
    (with D. Hundertmark, Y.-R. Lee, V. Zharnitsky)