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2. Multi-peak semiclassical bound states for Fractional Schrödinger Equations with fast decaying potentials.
- Author
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An, Xiaoming and Peng, Shuangjie
- Subjects
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SCHRODINGER equation , *MATHEMATICAL bounds , *PROBLEM solving , *MAXIMA & minima , *MATHEMATICAL formulas , *MATHEMATICAL models - Abstract
We study the following fractional Schrödinger equation where . Under some conditions on , we show that the problem has a family of solutions concentrating at any finite given local minima of provided that . All decay rates of are admissible. Especially, can be compactly supported. Different from the local case or the case of single-peak solutions, the nonlocal effect of the operator makes the peaks of the candidate solutions affect mutually, which causes more difficulties in finding solutions with multiple bumps. The methods in this paper are penalized technique and variational method. We study the following fractional Schrödinger equation \begin{equation*} \label{eq0.1} \varepsilon^{2s}(-\Delta)^s u + V(x)u = f(u), \,\,x\in\mathbb{R}^N, \end{equation*} where . Under some conditions on , we show that the problem has a family of solutions concentrating at any finite given local minima of provided that . All decay rates of are admissible. Especially, can be compactly supported. Different from the local case or the case of single-peak solutions, the nonlocal effect of the operator makes the peaks of the candidate solutions affect mutually, which causes more difficulties in finding solutions with multiple bumps. The methods in this paper are penalized technique and variational method. [ABSTRACT FROM AUTHOR]
- Published
- 2022
- Full Text
- View/download PDF
3. Some optical soliton solutions to the generalized (1 + 1)-dimensional perturbed nonlinear Schrödinger equation using two analytical approaches.
- Author
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Tariq, Kalim U., Seadawy, Aly R., Rizvi, Syed T. R., and Javed, Rizwan
- Subjects
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SCHRODINGER equation , *NONLINEAR Schrodinger equation , *MATHEMATICAL physics , *THEORY of wave motion , *HYPERBOLIC functions , *MATHEMATICAL models , *SOLITONS - Abstract
In this paper, we study the perturbed nonlinear Schrödinger equation (P-NLSE) representing the propagation of waves and may be regarded as a nonlinear complicated physical model. We use the modified extended Tan hyperbolic function (METhF) approach and the ( G ′ G 2 ) -expansion approach to construct some new traveling wave structures. Several solutions have been found including dark soliton, periodic-type solitons, bell-shaped solitons, single bell-shaped solitons. We also show a graphical representation of a number of exact solutions to the dynamical model together with a description of their behavior. The proposed techniques can also be extended to various nonlinear evolution models in mathematical physics and engineering. [ABSTRACT FROM AUTHOR]
- Published
- 2022
- Full Text
- View/download PDF
4. Nanoionics from a quantum mechanics point of view: Mathematical modeling and numerical simulation.
- Author
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Sepúlveda, Paulina, Muga, Ignacio, Sainz, Norberto, Rojas, René G., and Ossandón, Sebastián
- Subjects
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QUANTUM mechanics , *ELECTROMAGNETIC fields , *ELECTROMAGNETIC induction , *ELECTROMAGNETIC coupling , *MATHEMATICAL models , *TRP channels - Abstract
Solid nano-structures exhibiting fast ion transport (cations moving in anionic crystal structures) are becoming increasingly relevant in industrial applications. However, it is challenging to model their mechanics due to the presence of electromagnetic couplings. In this paper, a mathematical, physical, and computational framework is introduced, for a cation particle moving through an anion sub-lattice structure in the presence of two electromagnetic fields: an external electromagnetic field; and a self-induced electromagnetic field coming from back-reaction phenomena caused by the relative movement of cations with respect to the mentioned structure. Our approach seeks to incorporate magnetic effects, such as magnetic induction and spin of cations, which are not incorporated in other models, mainly due to the intrinsic difficulty of 3D effects. We propose a quantum mechanical formalism based on a Schrödinger-type equation, where a wave function models the behavior of a cation in presence of an external electromagnetic potential, coupled with transient and self-induced electromagnetic effects. To solve the model, a space–time coupled numerical scheme is presented, which allows the possibility of time-evolving electromagnetic effects. The technique uses finite-elements in space and time-marching schemes in time. While a time-explicit marching scheme is used to update the magnetic and electric-potential fields, a time-implicit marching scheme is used to solve the coupled Schrödinger equation. This strategy allows us to update the electromagnetic contributions and wave functions at each time-step. Numerical examples in one and two spatial dimensions (and evolving in time) have been implemented for some meaningful models obtained from nanoionics literature. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
- View/download PDF
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