Abstract
We use critical point theory to find solutions of the nonlinear steady state Schrödinger equations arising in the study of photonic lattices.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Similar content being viewed by others
References
G. Bartal, O. Manela, O. Cohen, J.W. Fleischer, M. Segev, Observation of second-band vortex solitons in 2D photonic lattices. Phys. Rev. Lett. 95, 053904 (2005)
S. Chen, Y. Lei, Existence of steady-state solutions in a nonlinear photonic lattice model. J. Math. Phys. 52(6), 063508 (2011)
W. Chen, D.L. Mills, Gap solitons and the nonlinear optical response of superlattices. Phys. Rev. Lett. 62, 1746–1749 (1989)
N.K. Efremidis, S. Sears, D.N. Christodoulides, Discrete solitons in photorefractive optically-induced photonic lattices. Phys.Rev.Lett. 85, 1863–1866 (2000)
W.J.W. Fleischer, M. Segev, N.K. Efremidis, D.N. Christodolides, Observation of two-dimensional discrete solitons in optically induced nonlinear photonic lattices. Nature 422(6928), 147–149 (2003)
J.W. Fleischer, G. Bartal, O. Cohen, O. Manela, M. Segev, J. Hudock, D.N. Christodoulides, Observation of vortex-ring discrete solitons in photonic lattices. Phys. Rev. Lett. 92, 123904 (2004)
P. Kuchment, The mathematics of photonic crystals, in Mathematical Modeling in Optical Science. Frontiers Application of Mathematical, vol. 22 (SIAM, Philadelphia, 2001), pp. 207–272
C. Liu, Q. Ren, On the steady-state solutions of a nonlinear photonic lattice model. J. Math. Phys. 56, 031501, 1–12 (2015). https://doi.org/10.1063/1.4914333
H. Martin, E.D. Eugenieva, Z. Chen, Discrete solitons and soliton-induced dislocations in partially coherent photonic lattices. Martin et al. Phys. Rev. Lett. 92, 123902 (2004)
D.N. Neshev, T.J. Alexander, E.A. Ostrovskaya, Y.S. Kivshar, H. Martin, I. Makasyuk, Z. Chen, Observation of discrete vortex solitons in optically induced photonic lattices. Phys. Rev. Lett. 92, 123903 (2004)
A. Pankov, Periodic nonlinear Schrödinger equation with application to photonic crystals. Milan J. Math. 73, 259–287 (2005)
M. Schechter, Linking Methods in Critical Point Theory (Birkhauser, Boston, 1999)
M. Schechter, An introduction to nonlinear analysis, in Cambridge Studies in Advanced Mathematics, vol. 95 (Cambridge University, Cambridge, 2004)
M. Schechter, The use of Cerami sequences in critical point theory theory. Abstr. Appl. Anal. 2007, 28 (2007). Art. ID 58948
M. Schechter, Minimax Systems and Critical Point Theory (Birkhauser, Boston, 2009)
M. Schechter, Steady state solutions for Schr/”odinger equations governing nonlinear optics. J. Math. Phys. 53, 043504, 8 pp. (2012)
M. Schechter, Photonic lattices. J. Math. Phys. 54, 061502, 7 pp. (2013)
M. Schechter, Critical Point Theory, Sandwich and Linking Systems (Birkhauser, Boston, 2020)
M. Schechter, Schrodinger equations in nonlinear optics, in Nonlinear Analysis and Global Optimization, ed. by Th. M. Rassias, P.M. Pardalos (Spriger, 2021), pp. 449–459
Y. Yang, Solition in Field Theory and Nonlinear Analysis (Springer, New York, 2001)
Y. Yang, R. Zhang, Steady state solutions for nonlinear Schrödinger equation arising in optics. J. Math. Phys. 50, 053501–053509 (2009)
J. Yang, A. Bezryadina, Z. Chen, I. Makasyuk, Observation of two-dimensional lattice vector solitons. Opt. Lett. 29, 1656 (2004)
J. Yang, I. Makasyuk, A. Bezryadina, Z. Chen, Dipole and quadrupole solitons in optically induced two-dimensional photonic lattices: theory and experiment. Studies Appl. Math. 113, 389–412 (2004)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2021 Springer Nature Switzerland AG
About this chapter
Cite this chapter
Schechter, M. (2021). Canonical Systems of Partial Differential Equations. In: Rassias, T.M. (eds) Nonlinear Analysis, Differential Equations, and Applications. Springer Optimization and Its Applications, vol 173. Springer, Cham. https://doi.org/10.1007/978-3-030-72563-1_22
Download citation
DOI: https://doi.org/10.1007/978-3-030-72563-1_22
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-72562-4
Online ISBN: 978-3-030-72563-1
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)