Classification of nonnegative solutions to static Schrödinger–Hartree–Maxwell system involving the fractional Laplacian

This paper is mainly concerned with the following semi-linear system involving the fractional Laplacian: $$ \textstyle\begin{cases} (-\Delta )^{\frac{\alpha }{2}}u(x)= (\frac{1}{ \vert \cdot \vert ^{\sigma }} \ast v^{p_{1}} )v^{p_{2}}(x), \quad x\in \mathbb{R}^{n}, \\ (-\Delta )^{\frac{\alpha }{2}}v(x)= (\frac{1}{ \vert \cdot \vert ^{\sigma }} \ast u^{q_{1}} )u^{q_{2}}(x), \quad x\in \mathbb{R}^{n}, \\ u(x)\geq 0,\quad\quad v(x)\geq 0, \quad x\in \mathbb{R}^{n}, \end{cases} $$ { ( − Δ ) α 2 u ( x ) = ( 1 | ⋅ | σ ∗ v p 1 ) v p 2 ( x ) , x ∈ R n , ( − Δ ) α 2 v ( x ) = ( 1 | ⋅ | σ ∗ u q 1 ) u q 2 ( x ) , x ∈ R n , u ( x ) ≥ 0 , v ( x ) ≥ 0 , x ∈ R n , where $0<\alpha \leq 2$ 0 < α ≤ 2 , $n\geq 2$ n ≥ 2 , $0<\sigma <n$ 0 < σ < n , and $0< p_{1}, q_{1}\leq \frac{2n-\sigma }{n-\alpha }$ 0 < p 1 , q 1 ≤ 2 n − σ n − α , $0< p_{2}, q_{2}\leq \frac{n+\alpha -\sigma }{n-\alpha }$ 0 < p 2 , q 2 ≤ n + α − σ n − α . Applying a variant (for nonlocal nonlinearity) of the direct method of moving spheres for fractional Laplacians, which was developed by W. Chen, Y. Li, and R. Zhang (J. Funct. Anal. 272(10):4131–4157, 2017), we derive the explicit forms for positive solution $(u,v)$ ( u , v ) in the critical case and nonexistence of positive solutions in the subcritical cases.

Discrete Nonlinear Schrödinger Systems for Periodic Media with Nonlocal Nonlinearity: The Case of Nematic Liquid Crystals

We study properties of an infinite system of discrete nonlinear Schrödinger equations that is equivalent to a coupled Schrödinger-elliptic differential equation with periodic coefficients. The differential equation was derived as a model for laser beam propagation in optical waveguide arrays in a nematic liquid crystal substrate and can be relevant to related systems with nonlocal nonlinearities. The infinite system is obtained by expanding the relevant physical quantities in a Wannier function basis associated to a periodic Schrödinger operator appearing in the problem. We show that the model can describe stable beams, and we estimate the optical power at different length scales. The main result of the paper is the Hamiltonian structure of the infinite system, assuming that the Wannier functions are real. We also give an explicit construction of real Wannier functions, and examine translation invariance properties of the linear part of the system in the Wannier basis.

Optical soliton perturbation and conservation law with Kudryashov’s refractive index having quadrupled power-law and dual form of generalized nonlocal nonlinearity

This paper is devoted to optical solitons for Kudryashov’s law of nonlinear refractive index, which stem from quadrupled-power law and dual form of nonlocal nonlinearity. The conservation law has been also exhibited to paint a complete picture of the model.