TY  - JOUR
T1  - Solution of a Nonlinear Schrödinger Equation with Galerkin’s Method
TT  - Lineer Olmayan bir Schrödinger Denkleminin Galerkin Metoduyla Çözümü
AU  - Yıldırım Aksoy, Nigar
PY  - 2017
DA  - June
Y2  - 2016
JF  - Journal of the Institute of Science and Technology
JO  - J. Inst. Sci. and Tech.
PB  - Igdir University
WT  - DergiPark
SN  - 2536-4618
SP  - 225
EP  - 239
VL  - 7
IS  - 2
LA  - en
AB  - In this paper, we consider an initial boundary value problem for a two-dimensional nonlinearSchrödinger equation. We prove by using Galerkin’s method that the solution of the initial boundary value problemexists and it has a unique solution. Also, we get an estimation for the solution of the initial boundary value problem.
KW  - Galerkin method
KW  - initial boundary value problem
KW  - Schrödinger equation
N2  - Bu çalışmada iki boyutlu lineer olmayan bir Schrödinger denklemi için bir başlangıç sınır değer problemigöz önüne alırız. Galerkin metodunu kullanarak başlangıç sınır değer probleminin çözümünün var ve tek olduğunuispatlarız. Ayrıca, başlangıç sınır değer probleminin çözümü için bir değerlendirme elde ederiz.
CR  - Bu C, 1994. An initial-buondary value problem of the nonlinear Schrödinger equation, Appl. Anal. 53: 241-254.
CR  - Bu C, Tsuyata K, Zhang C, 2005. Nonlinear Schrödinger equation with inhomogeneous Dirichlet-Boundary data. J. Math. Phys., 46: 083504.
CR  - Hashimoto H, Ono H, 1972. H. Nonlinear modulation of Gravity Waves. J. Phys. Soc. Jpn., 33: 805-811.
CR  - Holmer J, 2005. The initial-boundary value problem for the 1-d nonlinear Schrödinger equation on the half-line. Diff. Integ. Equation, 18: 647-668.
CR  - Hsieh P F, Sibuya Y, 1999. Basic Theory of Ordinary Differential Equations, Springer Verlag, New York. 468p.
CR  - Iskenderov A D, Yagubov G Y, 2007. Optimal control Problem with unbounded potential for multidimensional, nonlinear and nonstationary Schrödinger equation. Proceedings of the Lankaran State University, Natural Sciences series. 3-56.
CR  - Kaikina E I, 2013. Inhomogeneous Neumann initial-boundary value problem for the nonlinear                Schrödinger equation. Journal of Differential Equation, 255: 3338-3356.
CR  - Kelley P L,1965. Self-focusing of optical beams. Pyhsical Review Letters, 15: 1005-1008.
CR  - Ladyzhenskaya O A, 1985. The Boundary Value Problems of Mathematical Physics, Springer Verlag. 322p.
CR  - Mahmudov N M, 2007. Solvability of Boundary Value Problems for a Schrödinger Equation with Pure Imaginary Coefficient in the Nonlinear Part of This Equation. Proceedings of IMM of NAS of Azerbaijan, Vol.27, issue 35: 25-36.
CR  - Pontryagin L S, 1962. Ordinary Differential Equtions. Addison-Wesley Publishing Comp., (translated from the Russian).
CR  - Schimizu K, Ichikawa Y H, 1972. Automodulation of Ion Oscillation Modes in Plasma. J. Phys. Soc. Jpn., 33: 789-792.
CR  - Stewartson K, Stuart J T, 1971. A Nonlinear Instability Theory for a Wave System in Plane Poiseuille Flow. J. of Fluid Mechanic, 48(3): 529-545.
CR  - Strauss W, Bu C, 2001. An Inhomogeneous Boundary Value Problem for Nonlinear Schrödinger Equations. Journal of Differential Equations, 173: 79-91.
CR  - Talanov V I, 1965. Self-focusing of wave beams in nonlinear media. Soviet Physics-JETP Letters, 2:  138-141.
CR  - Tsutsumi M, 1991. On Global Solutions to the Initial Boundary Value Problem for Nonlinear Schrödinger Equation in Exterior Domain. Comm. Partial Diffential Equations, 6: 885-907.
CR  - Yildirim Aksoy N, Kocak Y, Ozeroglu Y, 2016. On the solvability of initial boundary value problems for nonlinear time-dependent Schrödinger equations. Quaestiones Mathematicae, 39(6): 751-771.
UR  - https://dergipark.org.tr/en/pub/jist/issue//389759
L1  - https://dergipark.org.tr/en/download/article-file/418491
ER  -