scholarly journals Two simple criterion to obtain exact controllability and stabilization of a linear family of dispersive PDE's on a periodic domain

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Francisco J. Vielma leal ◽  
Ademir Pastor
Keyword(s):  
Decay Rate ◽  
Fourth Order ◽  
Moment Method ◽  
Schrodinger Equation ◽  
Exact Controllability ◽  
Higher Order ◽  
Dispersive Equations ◽  
Schrödinger Equations ◽  
Periodic Domain ◽  
Simple Criterion

<p style='text-indent:20px;'>In this work, we use the classical moment method to find a practical and simple criterion to determine if a family of linearized Dispersive equations on a periodic domain is exactly controllable and exponentially stabilizable with any given decay rate in <inline-formula><tex-math id="M1">\begin{document}$ H_{p}^{s}(\mathbb{T}) $\end{document}</tex-math></inline-formula> with <inline-formula><tex-math id="M2">\begin{document}$ s\in \mathbb{R}. $\end{document}</tex-math></inline-formula> We apply these results to prove that the linearized Smith equation, the linearized dispersion-generalized Benjamin-Ono equation, the linearized fourth-order Schrödinger equation, and the Higher-order Schrödinger equations are exactly controllable and exponentially stabilizable with any given decay rate in <inline-formula><tex-math id="M3">\begin{document}$ H_{p}^{s}(\mathbb{T}) $\end{document}</tex-math></inline-formula> with <inline-formula><tex-math id="M4">\begin{document}$ s\in \mathbb{R}. $\end{document}</tex-math></inline-formula></p>

scholarly journals Schrödinger equations in noncylindrical domains: exact controllability

2006 ◽  
Vol 2006 ◽  
pp. 1-29
Author(s):  
G. O. Antunes ◽  
M. D. G. da Silva ◽  
R. F. Apolaya
Keyword(s):  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Exact Controllability ◽  
Schrodinger Equations ◽  
Lateral Boundary ◽  
Schrödinger Equations ◽  
Orthogonal Matrices ◽  
Noncylindrical Domain ◽  
Bounded Set

We consider an open bounded setΩ⊂ℝnand a family{K(t)}t≥0of orthogonal matrices ofℝn. SetΩt={x∈ℝn;x=K(t)y,for all y∈Ω}, whose boundary isΓt. We denote byQ^the noncylindrical domain given byQ^=∪0<t<T{Ωt×{t}}, with the regular lateral boundaryΣ^=∪0<t<T{Γt×{t}}. In this paper we investigate the boundary exact controllability for the linear Schrödinger equationu′−iΔu=finQ^(i2=−1),u=wonΣ^,u(x,0)=u0(x)inΩ0, wherewis the control.

An inverse boundary problem for fourth-order Schrödinger equations with partial data

2019 ◽  
Vol 69 (1) ◽  
pp. 125-138
Author(s):  
Zhiwen Duan ◽  
Shuxia Han
Keyword(s):  
Boundary Problem ◽  
Fourth Order ◽  
Schrodinger Equation ◽  
Geometrical Optics ◽  
Cauchy Data ◽  
Carleman Estimates ◽  
Schrödinger Equations ◽  
Inverse Boundary ◽  
Partial Data ◽  
Inverse Boundary Problem

Abstract In this paper, we show that in dimension n ≥ 3, the knowledge of the Cauchy data for the fourth-order Schrödinger equation measured on possibly very small subsets of the boundary determines uniquely the potential. The proof is based on the Carleman estimates and the construction of complex geometrical optics solutions.

Oscillations in the Interactions Among Multiple Solitons in an Optical Fibre

2016 ◽  
Vol 71 (12) ◽  
pp. 1079-1091 ◽  
Author(s):  
Wen-Qiang Hu ◽  
Yi-Tian Gao ◽  
Chen Zhao ◽  
Yu-Jie Feng ◽  
Chuan-Qi Su
Keyword(s):  
Optical Fibre ◽  
Fourth Order ◽  
Schrodinger Equation ◽  
Higher Order ◽  
Nonlinear Schrodinger Equation ◽  
Order Operator ◽  
Eighth Order ◽  
Spectral Parameters ◽  
The Third ◽  
Seventh Order

AbstractIn this article, under the investigation on the interactions among multiple solitons for an eighth-order nonlinear Schrödinger equation in an optical fibre, oscillations in the interaction zones are observed theoretically. With different coefficients of the operators in this equation, we find that (1) the oscillations in the solitonic interaction zones have different forms with different spectral parameters of this equation; (2) the oscillations in the interactions among the multiple solitons are affected by the choice of spectral parameters, the dispersive effects and nonlinearity of the eighth-order operator; (3) the second-, fifth-, sixth-, and seventh-order operators restrain oscillations in the solitonic interaction zones and the higher-order operators have stronger attenuated effects than the lower ones, while the third- and fourth-order operators stimulate and extend the scope of oscillations.

Analytical methods via bright–dark solitons and solitary wave solutions of the higher-order nonlinear Schrödinger equation with fourth-order dispersion

2019 ◽  
Vol 33 (35) ◽  
pp. 1950443 ◽  
Author(s):  
Aly R. Seadawy ◽  
Mujahid Iqbal ◽  
Dianchen Lu
Keyword(s):  
Solitary Wave ◽  
Fourth Order ◽  
Schrodinger Equation ◽  
Higher Order ◽  
Nonlinear Schrodinger Equation ◽  
Solitary Wave Solutions ◽  
Physical Sciences ◽  
Dark Solitons ◽  
Wave Solutions ◽  
Order Dispersion

In this research work, we investigated the higher-order nonlinear Schrödinger equation (NLSE) with fourth-order dispersion, self-steepening, nonlinearity, nonlinear dispersive terms and cubic-quintic terms which is described as the propagation of ultra-short pulses in fiber optics. We apply the modification form of extended auxiliary equation mapping method to find the new exact and solitary wave solutions of higher-order NLSE. As a result, new solutions are obtained in the form of solitons, kink–anti-kink type solitons, bright–dark solitons, traveling wave, trigonometric functions, elliptic functions and periodic solitary wave solutions. These new different types of solutions show the power and fruitfulness of this new method and also show two- and three-dimensional graphically with the help of computer software Mathematica. These new solutions have many applications in the field of physics and other branches of physical sciences. We can also solve other higher-order nonlinear partial differential equations (NPDEs) involved in mathematical physics and other various branches of physical sciences with this new technique.

Modulational Instability of the Higher-Order Nonlinear Schrödinger Equation with Fourth-Order Dispersion and Quintic Nonlinear Terms

2006 ◽  
Vol 61 (5-6) ◽  
pp. 225-234 ◽  
Author(s):  
Woo-Pyo Hong
Keyword(s):  
Schrödinger Equation ◽  
Nonlinear Schrödinger Equation ◽  
Fourth Order ◽  
Schrodinger Equation ◽  
Modulational Instability ◽  
Higher Order ◽  
Nonlinear Schrodinger Equation ◽  
Normal Dispersion ◽  
Nonlinear Schrödinger ◽  
Order Dispersion

The modulational instability of the higher-order nonlinear Schrödinger equation with fourth-order dispersion and quintic nonlinear terms, describing the propagation of extremely short pulses, is investigated. Several types of gains by modulational instability are shown to exist in both the anomalous and normal dispersion regimes depending on the sign and strength of the higher-order nonlinear terms. The evolution of the modulational instability in both the anomalous and normal dispersion regimes is numerically investigated and the effects of the higher-order dispersion and nonlinear terms on the formation and evolution of the solitons induced by modulational instability are studied. - PACS numbers: 42.65.Tg, 42.81Dp, 42.65Sf

scholarly journals Solitary Wave-Series Solutions to Non-Linear Schrodinger Equations

2012 ◽  
Vol 13 (1) ◽  
Author(s):  
Muhammad Azram ◽  
H. Zaman
Keyword(s):  
Solitary Wave ◽  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Series Solution ◽  
Higher Order ◽  
Schrodinger Equations ◽  
Schrödinger Equations ◽  
Series Solutions ◽  
Non Linear ◽  
Continuity And Discontinuity

ABSTRACT: In this paper, higher-order dispersive non-linear Schrodinger equations are studied. Their solitary wave-series solutions with continuity of the derivatives and specific discontinuity of the derivatives at the crest are presented. Furthermore, convergence of the series’ solutions is also validated and discussed with the help of graphs. ABSTRAK: Kertas ini mengkaji persamaan Schrodinger serakan taklinear turutan tinggi. Penyelesaian siri-gelombang tunggalnya dengan kamiran berterusan dan kamiran tak berterusan pada maksimum telah dibentangkan. Penumpuan penyelesaian siri juga telah diperiksa dan dibincangkan dengan bantuan graf-graf.KEYWORDS: Schrodinger equation; solitary wave-series solution; continuity and discontinuity of derivatives at crest

scholarly journals Dynamics of the Optical Pulse in a Nonlinear Medium: Approach of Moment Method Coupled with the Fourth Order Runge-Kutta Method

2020 ◽  
pp. 8-17
Author(s):  
Fessomon Koki ◽  
Gaston Edah ◽  
Minadohona Maxime Capo- Chichi ◽  
Gaetan Finagnon Djossou ´ ◽  
Camille Elloh ◽  
...  
Keyword(s):  
Fourth Order ◽  
Moment Method ◽  
Nonlinear Medium ◽  
Schrodinger Equation ◽  
Nonlinear Schrodinger Equation ◽  
Kutta Method ◽  
Nonlinear Dispersion ◽  
Runge Kutta Method ◽  
Runge Kutta ◽  
The Moment

In this paper, we considered the nonlinear Schrodinger equation and applied the moment method ¨ in order to investigate the evolution of pulse parameters in nonlinear medium. This mathematical model described the effects of cubic nonlinear and the nonlinear dispersion terms on the soliton.  The application of the moment method leads to variational equations that is integrated numerically by the fourth order Runge-Kutta method. The results obtained shows the variations of some important parameters of the pulse namely the energy, the pulse position, the frequency shift, the chirp and the width. It reveals the effects of the nonlinear dispersion and nonlinear cubic terms on each parameter on the pulse. The moment method is appropriate to study the dynamics of theoptical pulse in a nonlinear medium modelled by the nonlinear Schrodinger equation.

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