Solitons propagation dynamics in a saturable PT-symmetric fractional medium

In the current research, the propagation of solitons in a saturable PT-symmetric fractional system is studied by solving nonlinear fractional Schrödinger equation. Three numerical methods are employed for this purpose, namely Monte Carlo based Euler-Lagrange variational schema, split-step method, and extrapolation approach. The results show good agreement and accuracy. The effect of different parameters such as potential depth, Levy indices, and saturation parameter, on the physical properties of the systems such as maximum intensity and soliton width oscillations are considered.

Nonlinear perturbations of a periodic fractional Laplacian with supercritical growth

Our main goal is to explore the existence of positive solutions for a class of nonlinear fractional Schrödinger equation involving supercritical growth given by $$ (- \Delta)^{\alpha} u + V(x)u=p(u),\quad x\in \mathbb{R^N},\ N \geq 1. $$ We analyze two types of problems, with $V$ being periodic and asymptotically periodic; for this we use a variational method, a truncation argument and a concentration compactness principle.

Existence of Ground States of Fractional Schrödinger Equations

We consider ground states of the nonlinear fractional Schrödinger equation with potentials ( - Δ ) s ⁢ u + V ⁢ ( x ) ⁢ u = f ⁢ ( x , u ) , s ∈ ( 0 , 1 ) , (-\Delta)^{s}u+V(x)u=f(x,u),\quad s\in(0,1), on the whole space ℝ N {\mathbb{R}^{N}} , where V is a periodic non-negative nontrivial function on ℝ N {\mathbb{R}^{N}} and the nonlinear term f has some proper growth on u. Under uniform bounded assumptions about V, we can show the existence of a ground state. We extend the result of Li, Wang, and Zeng to the fractional case.

Normalized solutions of supercritical nonlinear fractional Schrödinger equation with potential

<p style='text-indent:20px;'>We are concerned with the following nonlinear fractional Schrödinger equation:</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$\begin{equation} (-\Delta)^s u+V(x)u+\omega u = |u|^{p-2}u\quad {\rm{in}}\,\,{\mathbb{R}}^N,\;\;\;\;\;\;({\textbf{P}})\end{equation}$ \end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>where <inline-formula><tex-math id="M1">\begin{document}$ s\in(0,1) $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M2">\begin{document}$ p\in\left(2+4s/N,2^*_s\right) $\end{document}</tex-math></inline-formula>, that is, the mass supercritical and Sobolev subcritical. Under certain assumptions on the potential <inline-formula><tex-math id="M3">\begin{document}$ V:{\mathbb{R}}^N\rightarrow {\mathbb{R}} $\end{document}</tex-math></inline-formula>, positive and vanishing at infinity including potentials with singularities (which is important for physical reasons), we prove that there exists at least one <inline-formula><tex-math id="M4">\begin{document}$ L^2 $\end{document}</tex-math></inline-formula>-normalized solution <inline-formula><tex-math id="M5">\begin{document}$ (u,\omega)\in H^s({\mathbb{R}}^N)\times{\mathbb{R}}^+ $\end{document}</tex-math></inline-formula> of equation (P). In order to overcome the lack of compactness, the proof is based on a new min-max argument and splitting lemma for nonlocal version.</p>