• Previous Article
    Solvability of generalized nonlinear heat equations with constraints coupled with Navier--Stokes equations in 2D domains
  • PROC Home
  • This Issue
  • Next Article
    Simulation of complex dynamics using POD 'on the fly' and residual estimates
2015, 2015(special): 1070-1078. doi: 10.3934/proc.2015.1070

A new way for decreasing of amplitude of wave reflected from artificial boundary condition for 1D nonlinear Schrödinger equation

1. 

Lomonosov Moscow State University, Leninskie Gory, Moscow 119992, Russian Federation, Russian Federation

Received  September 2014 Revised  January 2015 Published  November 2015

In this report, we focus on computation performance enhancing at computer simulation of a laser pulse interaction with optical periodic structure (photonic crystal). Decreasing the domain before the photonic crystal one can essentially increases a computation performance. With this aim, firstly, we provide a computation in linear medium and storage the complex amplitude at chosen section of coordinate along which a laser light propagates. Then we use this time-dependent value of complex amplitude as left boundary condition for the 1D nonlinear Schrödinger equation in decreased domain containing a nonlinear photonic crystal. Because a part of a laser pulse reflects from faces of photonic crystal layers, we use artificial boundary condition. To decrease amplitude of the wave reflected from artificial boundary we introduce in consideration some additional number of waves related with the equation under consideration.
Citation: Vyacheslav A. Trofimov, Evgeny M. Trykin. A new way for decreasing of amplitude of wave reflected from artificial boundary condition for 1D nonlinear Schrödinger equation. Conference Publications, 2015, 2015 (special) : 1070-1078. doi: 10.3934/proc.2015.1070
References:
[1]

X. Antoine, C. Besse and V. Mouysset, Numerical schemes for the simulation of the two-dimensional Schrödinger equation using non-reflecting boundary conditions, Math Comput, 73 (2004), 1779-1799. Google Scholar

[2]

E.B. Tereshin, V.A. Trofimov and M.V.Fedotov, Conservative finite difference scheme for the problem of propagation of a femtosecond pulse in a nonlinear photonic crystal with non-reflecting boundary conditions, Comp Math and Math Physics, 46:1 (2006), 154-164. Google Scholar

[3]

V.A. Trofimov and P.V. Dogadushkin, Boundary conditions for the problem of femtosecond pulse propagation in absorption layered structure, Proceedings of Fourth Int Conf Finite Diff Methods: Theory and app, Rouse University Angel Kanchev, Lozenetz, Bulgaria, (2007), 307-313. Google Scholar

[4]

Z. Xuand and H. Han, Absorbing boundary conditions for nonlinear Schrödinger equations, Phys Rev, 74:037704 (2006). Google Scholar

[5]

C. Zheng, Exact nonreflecting boundary conditions for one-dimensional cubic nonlinear Schrödinger equations, J Comput Phys, 215 (2006), 552-565. Google Scholar

show all references

References:
[1]

X. Antoine, C. Besse and V. Mouysset, Numerical schemes for the simulation of the two-dimensional Schrödinger equation using non-reflecting boundary conditions, Math Comput, 73 (2004), 1779-1799. Google Scholar

[2]

E.B. Tereshin, V.A. Trofimov and M.V.Fedotov, Conservative finite difference scheme for the problem of propagation of a femtosecond pulse in a nonlinear photonic crystal with non-reflecting boundary conditions, Comp Math and Math Physics, 46:1 (2006), 154-164. Google Scholar

[3]

V.A. Trofimov and P.V. Dogadushkin, Boundary conditions for the problem of femtosecond pulse propagation in absorption layered structure, Proceedings of Fourth Int Conf Finite Diff Methods: Theory and app, Rouse University Angel Kanchev, Lozenetz, Bulgaria, (2007), 307-313. Google Scholar

[4]

Z. Xuand and H. Han, Absorbing boundary conditions for nonlinear Schrödinger equations, Phys Rev, 74:037704 (2006). Google Scholar

[5]

C. Zheng, Exact nonreflecting boundary conditions for one-dimensional cubic nonlinear Schrödinger equations, J Comput Phys, 215 (2006), 552-565. Google Scholar

[1]

Abderrazak Chrifi, Mostafa Abounouh, Hassan Al Moatassime. Galerkin method of weakly damped cubic nonlinear Schrödinger with Dirac impurity, and artificial boundary condition in a half-line. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021030

[2]

M. D. Todorov, C. I. Christov. Conservative numerical scheme in complex arithmetic for coupled nonlinear Schrödinger equations. Conference Publications, 2007, 2007 (Special) : 982-992. doi: 10.3934/proc.2007.2007.982

[3]

Bernard Ducomet, Alexander Zlotnik, Ilya Zlotnik. On a family of finite-difference schemes with approximate transparent boundary conditions for a generalized 1D Schrödinger equation. Kinetic & Related Models, 2009, 2 (1) : 151-179. doi: 10.3934/krm.2009.2.151

[4]

Minoru Murai, Kunimochi Sakamoto, Shoji Yotsutani. Representation formula for traveling waves to a derivative nonlinear Schrödinger equation with the periodic boundary condition. Conference Publications, 2015, 2015 (special) : 878-900. doi: 10.3934/proc.2015.0878

[5]

Frank Wusterhausen. Schrödinger equation with noise on the boundary. Conference Publications, 2013, 2013 (special) : 791-796. doi: 10.3934/proc.2013.2013.791

[6]

Makoto Okumura, Daisuke Furihata. A structure-preserving scheme for the Allen–Cahn equation with a dynamic boundary condition. Discrete & Continuous Dynamical Systems, 2020, 40 (8) : 4927-4960. doi: 10.3934/dcds.2020206

[7]

Hans Zwart, Yann Le Gorrec, Bernhard Maschke. Relating systems properties of the wave and the Schrödinger equation. Evolution Equations & Control Theory, 2015, 4 (2) : 233-240. doi: 10.3934/eect.2015.4.233

[8]

Hongwei Zhang, Qingying Hu. Asymptotic behavior and nonexistence of wave equation with nonlinear boundary condition. Communications on Pure & Applied Analysis, 2005, 4 (4) : 861-869. doi: 10.3934/cpaa.2005.4.861

[9]

Muhammad I. Mustafa. On the control of the wave equation by memory-type boundary condition. Discrete & Continuous Dynamical Systems, 2015, 35 (3) : 1179-1192. doi: 10.3934/dcds.2015.35.1179

[10]

Alain Hertzog, Antoine Mondoloni. Existence of a weak solution for a quasilinear wave equation with boundary condition. Communications on Pure & Applied Analysis, 2002, 1 (2) : 191-219. doi: 10.3934/cpaa.2002.1.191

[11]

M. Eller. On boundary regularity of solutions to Maxwell's equations with a homogeneous conservative boundary condition. Discrete & Continuous Dynamical Systems - S, 2009, 2 (3) : 473-481. doi: 10.3934/dcdss.2009.2.473

[12]

Lassaad Aloui, Moez Khenissi. Boundary stabilization of the wave and Schrödinger equations in exterior domains. Discrete & Continuous Dynamical Systems, 2010, 27 (3) : 919-934. doi: 10.3934/dcds.2010.27.919

[13]

Gianluca Frasca-Caccia, Peter E. Hydon. Locally conservative finite difference schemes for the modified KdV equation. Journal of Computational Dynamics, 2019, 6 (2) : 307-323. doi: 10.3934/jcd.2019015

[14]

Xiaorui Wang, Genqi Xu, Hao Chen. Uniform stabilization of 1-D Schrödinger equation with internal difference-type control. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021022

[15]

Umberto De Maio, Akamabadath K. Nandakumaran, Carmen Perugia. Exact internal controllability for the wave equation in a domain with oscillating boundary with Neumann boundary condition. Evolution Equations & Control Theory, 2015, 4 (3) : 325-346. doi: 10.3934/eect.2015.4.325

[16]

Youngmok Jeon, Dongwook Shin. Immersed hybrid difference methods for elliptic boundary value problems by artificial interface conditions. Electronic Research Archive, , () : -. doi: 10.3934/era.2021043

[17]

Haoyue Cui, Dongyi Liu, Genqi Xu. Asymptotic behavior of a Schrödinger equation under a constrained boundary feedback. Mathematical Control & Related Fields, 2018, 8 (2) : 383-395. doi: 10.3934/mcrf.2018015

[18]

Phan Van Tin. On the Cauchy problem for a derivative nonlinear Schrödinger equation with nonvanishing boundary conditions. Evolution Equations & Control Theory, 2021  doi: 10.3934/eect.2021028

[19]

Satoshi Masaki. A sharp scattering condition for focusing mass-subcritical nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, 2015, 14 (4) : 1481-1531. doi: 10.3934/cpaa.2015.14.1481

[20]

Tomáš Bodnár, Philippe Fraunié, Petr Knobloch, Hynek Řezníček. Numerical evaluation of artificial boundary condition for wall-bounded stably stratified flows. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 785-801. doi: 10.3934/dcdss.2020333

 Impact Factor: 

Metrics

  • PDF downloads (80)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]