Periodic Nonlinear Schrödinger Equation with Application to Photonic Crystals

By Alexander Pankov. 2004

343 citations recorded by the source corpus.

Abstract.We present some results on existence of nontrivial solutions of periodic stationary nonlinear Schrödinger equations. We also sketch an application to nonlinear optics and discuss some open problems.

Subfields: Applied Mathematics, Computational Theory and Mathematics, Numerical Analysis

View Paper: https://www.semanticscholar.org/paper/1127c96b4b9a830c84d3def880f1fe39a3f4dbbe

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In Applied Mathematics

In Computational Theory and Mathematics

In Numerical Analysis

Citations in the corpus

New Super-quadratic Conditions on Ground State Solutions for Superlinear Schrödinger Equation (2014, neutral): “(SQ) was first introduced by Liu and Wang [12] and then was commonly used in recent papers, see [4, 5, 6, 8, 9, 13, 14, 16, 20, 18, 23].”
Hamiltonian elliptic systems: a guide to variational frameworks (2014, methodology): “The work [98] is inspired by Pankov [76], where a manifold of type (5.4) appeared in an independent way.”
NON-NEHARI-MANIFOLD METHOD FOR ASYMPTOTICALLY LINEAR SCHRÖDINGER EQUATION (2014, neutral): “In [10], based on the Nehari-manifold method, Szulkin and Weth developed an approach to ﬁnd ground state solutions of Nehari–Pankov type for problem (1.1) with superlinear term f .”
NON-NEHARI MANIFOLD METHOD FOR SUPERLINEAR SCHRÖDINGER EQUATION (2014, background): “Unlike the Nehari manifold method, our approach lies on finding a minimizing Cerami sequence for functional Φ outside the Nehari-Pankov manifold N− by using the diagonal method, see Lemma 3.8.”
Non-Nehari manifold method for asymptotically periodic Schrödinger equations (2014, methodology): “The set N− was first introduced by Pankov [20], which is a subset of the Nehari manifold N = { u ∈ E \ {0} : 〈Φ′(u), u〉 = 0 }”
New conditions on nonlinearity for a periodic Schrödinger equation having zero as spectrum (2014, background): “There is a large literature on existence and multiplicity results, for example, [3,5–7,9–11,15–17,19–23,27,28,30,32,33].”
The method of Nehari manifold (2010, methodology): “In order to circumvent this difficulty we shall replace N by the generalized Nehari manifold M which has been introduced by Pankov in [35].”