Periodic Nonlinear Schrödinger Equation with Application to Photonic Crystals
TLDR
In this article, the existence of nontrivial solutions of periodic stationary nonlinear Schrodinger equations has been studied and an application to nonlinear optics and open problems have been discussed.Abstract:
We present some results on existence of nontrivial solutions of periodic stationary nonlinear Schrodinger equations. We also sketch an application to nonlinear optics and discuss some open problems.read more
Citations
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Ground state solutions for some indefinite variational problems
Andrzej Szulkin,Tobias Weth +1 more
TL;DR: In this article, Li et al. developed an approach to find ground state solutions, i.e., nontrivial solutions with least possible energy, based on a direct reduction of the indefinite variational problem to a definite one.
The method of Nehari manifold
Andrzej Szulkin,Tobias Weth +1 more
TL;DR: A unified approach to the method of Nehari manifold for functionals which have a local minimum at 0 is presented in this article, where several examples where this method is applied to the problem of finding ground states and multiple solutions for nonlinear elliptic boundary value problems.
Journal ArticleDOI
Non-Nehari manifold method for asymptotically periodic Schrödinger equations
TL;DR: Li et al. as discussed by the authors developed a more direct approach to generalize the main result of Szulkin and Weth by removing the strictly increasing condition in the Nehari type assumption on f(x, t)/|t|.
Book
Localization in Periodic Potentials: From Schrodinger Operators to the Gross-Pitaevskii Equation
TL;DR: In this paper, the authors provide a comprehensive treatment of the Gross-Pitaevskii equation with a periodic potential; in particular, the localized modes supported by the periodic potential.
Journal ArticleDOI
Gap solitons in periodic discrete nonlinear Schrödinger equations
TL;DR: In this paper, it was shown that the periodic discrete nonlinear Schrodinger equation with cubic nonlinearity possesses gap solutions, i.e., standing waves with the frequency in a spectral gap, that are exponentially localized in the spatial variable.
References
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Book
Photonic Crystals: Molding the Flow of Light
TL;DR: In this paper, the authors developed the theoretical tools of photonics using principles of linear algebra and symmetry, emphasizing analogies with traditional solid-state physics and quantum theory, and investigated the unique phenomena that take place within photonic crystals at defect sites and surfaces, from one to three dimensions.
Book
Minimax methods in critical point theory with applications to differential equations
TL;DR: The mountain pass theorem and its application in Hamiltonian systems can be found in this paper, where the saddle point theorem is extended to the case of symmetric functionals with symmetries and index theorems.
Journal ArticleDOI
The concentration-compactness principle in the calculus of variations. The locally compact case, part 1
TL;DR: In this paper, the equivalence between the compactness of all minimizing sequences and some strict sub-additivity conditions was shown based on a compactness lemma obtained with the help of the notion of concentration function of a measure.