Solitary waves in nonlocal NLS with dispersion averaged saturated nonlinearities. (English) Zbl 1395.35173

Hundertmark, Dirk; Lee, Young-Ran; Ried, Tobias; Zharnitsky, Vadim

J. Differ. Equations 265, No. 8, 3311-3338 (2018).

Authors’ abstract: A nonlinear Schrödinger equation (NLS) with dispersion averaged nonlinearity of saturated type is considered. Such a nonlocal NLS is of integro-differential type and it arises naturally in modeling fiber-optics communication systems with periodically varying dispersion profile (dispersion management). The associated constrained variational principle is shown to posses a ground state solution by constructing a convergent minimizing sequence through the application of a method similar to the classical concentration compactness principle of Lions. One of the obstacles in applying this variational approach is that a saturated nonlocal nonlinearity does not satisfy uniformly the so-called strict sub-additivity condition. This is overcome by applying a special version of Ekeland’s variational principle.

Reviewer: Alessandro Selvitella (Ottawa)

MSC:

35Q55	NLS equations (nonlinear Schrödinger equations)
35A15	Variational methods applied to PDEs
35R09	Integro-partial differential equations

Keywords:

saturated nonlinearities; nonlocal NLS; solitary waves; nonlocal variational problems

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