Lineer Olmayan bir Schrödinger Denkleminin Galerkin Metoduyla Çözümü

Abstract

Bu çalışmada iki boyutlu lineer olmayan bir Schrödinger denklemi için bir başlangıç sınır değer problemi
göz önüne alırız. Galerkin metodunu kullanarak başlangıç sınır değer probleminin çözümünün var ve tek olduğunu
ispatlarız. Ayrıca, başlangıç sınır değer probleminin çözümü için bir değerlendirme elde ederiz.

Keywords

• Başlangıç sınır değer problemi,Galerkin metodu,Schrödinger denklemi

Solution of a Nonlinear Schrödinger Equation with Galerkin’s Method

Abstract

In this paper, we consider an initial boundary value problem for a two-dimensional nonlinear
Schrödinger equation. We prove by using Galerkin’s method that the solution of the initial boundary value problem
exists and it has a unique solution. Also, we get an estimation for the solution of the initial boundary value problem.

Keywords

• Galerkin method,initial boundary value problem,Schrödinger equation

References

1. Bu C, 1994. An initial-buondary value problem of the nonlinear Schrödinger equation, Appl. Anal. 53: 241-254.
2. Bu C, Tsuyata K, Zhang C, 2005. Nonlinear Schrödinger equation with inhomogeneous Dirichlet-Boundary data. J. Math. Phys., 46: 083504.
3. Hashimoto H, Ono H, 1972. H. Nonlinear modulation of Gravity Waves. J. Phys. Soc. Jpn., 33: 805-811.
4. 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.
5. Hsieh P F, Sibuya Y, 1999. Basic Theory of Ordinary Differential Equations, Springer Verlag, New York. 468p.
6. 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.
7. Kaikina E I, 2013. Inhomogeneous Neumann initial-boundary value problem for the nonlinear Schrödinger equation. Journal of Differential Equation, 255: 3338-3356.
8. Kelley P L,1965. Self-focusing of optical beams. Pyhsical Review Letters, 15: 1005-1008.

9. Ladyzhenskaya O A, 1985. The Boundary Value Problems of Mathematical Physics, Springer Verlag. 322p.
10. 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.
11. Pontryagin L S, 1962. Ordinary Differential Equtions. Addison-Wesley Publishing Comp., (translated from the Russian).
12. Schimizu K, Ichikawa Y H, 1972. Automodulation of Ion Oscillation Modes in Plasma. J. Phys. Soc. Jpn., 33: 789-792.
13. 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.
14. Strauss W, Bu C, 2001. An Inhomogeneous Boundary Value Problem for Nonlinear Schrödinger Equations. Journal of Differential Equations, 173: 79-91.
15. Talanov V I, 1965. Self-focusing of wave beams in nonlinear media. Soviet Physics-JETP Letters, 2: 138-141.
16. 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.
17. 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.