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

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.


35Q55 NLS equations (nonlinear Schrödinger equations)
35A15 Variational methods applied to PDEs
35R09 Integro-partial differential equations
Full Text: DOI arXiv Link


[1] Ablowitz, M. J.; Biondini, G., Multiscale pulse dynamics in communication systems with strong dispersion management, Opt. Lett., 23, 1668-1670, (1998)
[2] Ambrosetti, A.; Rabinowitz, P. H., Dual variational methods in critical point theory and applications, J. Funct. Anal., 14, 349-381, (1973), MR 0370183. Zbl 0273.49063 · Zbl 0273.49063
[3] Choi, M.-R.; Hundertmark, D.; Lee, Y.-R., Thresholds for existence of dispersion management solitons for general nonlinearities, SIAM J. Math. Anal., 49, 1519-1569, (2017), MR 3639574 · Zbl 1432.35184
[4] Costa, D. G., An invitation to variational methods in differential equations, (2007), Birkhäuser Boston, MR 2321283. Zbl 1123.35001 · Zbl 1123.35001
[5] de Almeida Maia, L.; Montefusco, E.; Pellacci, B., Weakly coupled nonlinear Schrödinger systems: the saturation effect, Calc. Var. Partial Differential Equations, 46, 325-351, (2013), MR 3016511. Zbl 1257.35171 · Zbl 1257.35171
[6] Ekeland, I., On the variational principle, J. Math. Anal. Appl., 47, 324-353, (1974), MR 0346619. Zbl 0286.49015 · Zbl 0286.49015
[7] Gabitov, I. R.; Turitsyn, S. K., Averaged pulse dynamics in a cascaded transmission system with passive dispersion compensation, Opt. Lett., 21, 327-329, (1996)
[8] Gatz, S.; Herrmann, J., Soliton propagation in materials with saturable nonlinearity, J. Opt. Soc. Amer. B, 8, 2296-2302, (1991)
[9] Hundertmark, D.; Lee, Y.-R., On non-local variational problems with lack of compactness related to non-linear optics, J. Nonlinear Sci., 22, 1-38, (2012), MR 2878650. Zbl 1244.49008 · Zbl 1244.49008
[10] Hundertmark, D.; Lee, Y.-R.; Ried, T.; Zharnitsky, V., Dispersion managed solitons in the presence of saturated nonlinearity, Phys. D, (2017) · Zbl 1378.35279
[11] Jeanjean, L.; Tanaka, K., A positive solution for an asymptotically linear elliptic problem on \(\mathbb{R}^N\) autonomous at infinity, ESAIM Control Optim. Calc. Var., 7, 597-614, (2002), MR 1925042. Zbl 1225.35088 · Zbl 1225.35088
[12] Kunze, M., On a variational problem with lack of compactness related to the Strichartz inequality, Calc. Var. Partial Differential Equations, 19, 307-336, (2004), MR 2033144 · Zbl 1352.49002
[13] Mandel, R., Minimal energy solutions and infinitely many bifurcating branches for a class of saturated nonlinear Schrödinger systems, Adv. Nonlinear Stud., 16, 95-113, (2016), MR 3456750. Zbl 1334.35319 · Zbl 1334.35319
[14] Usman, A.; Osman, J.; Tilley, D. R., Analytical solitary wave solutions for nonlinear Schrödinger equations with denominating saturating nonlinearity, J. Phys. A: Math. Gen., 31, 8397-8403, (1998), Zbl 0986.81025 · Zbl 0986.81025
[15] Zharnitsky, V.; Grenier, E.; Jones, C. K.R. T.; Turitsyn, S. K., Stabilizing effects of dispersion management, Phys. D, 152-153, 794-817, (2001), MR 837940. Zbl 0972.78021 · Zbl 0972.78021
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.