Nonlineer Schrödinger Denkleminin Tam Çözümleri

Yıl 2019, Cilt: 2 Sayı: 1, 11 - 19, 02.05.2019

Halide Gümüş Halis Yılmaz

Öz

Bu makale Nonlineer Schrödinger denklemine ayrılmıştır. Öncelikle denklem hakkında bilgiler verilmiş ve daha sonra denklem enerji korumalı yöntem ile elde edilmiştir. Sonraki bölümde de ilerleyen dalga çözümünü elde ettik ve daha sonra Hirota metodunu kullanarak NLS denkleminin 1-soliton ve 2-soliton çözümünü verdik. Sonuç ve tartışma kısmı da beşinci bölümde verilmiştir.

Anahtar Kelimeler

İlerleyen dalga, Nonlineer Schrödinger (NLS) Denklemi, Hirota metodu

Exact Solutions of Nonlinear Schrödinger Equation

Öz

This article is devoted to NLS Equation. Firstly, we give some information about NLS then we obtain the equation by energy preserving method. Furthermore we construct a travelling wave solution for the NLS equation. Finally, by using Hirota's direct method we present one-soliton and two soliton solutions for the NLS.

Anahtar Kelimeler

Travelling wave, Nonlineer Schrödinger (NLS) Equation, Hirota's Direct Method

Kaynakça

• Ablowitz, M. J., Ladik, J. F. 1975. Nonlinear differential-difference equations. Journal of indent Mathematical Physics, 16, 598-603.
• Hirota , R. 1980. Direct methods in soliton thoery.Solitons. Topics in Current Physics, vol. 17. Berlin:Springer-Verlag.
• Hirota, R. 2004. The Direct Method in Soliton Theory.New York: Cambridge University Press, 12-58.
• Hirota, R. 1971. Exact Solution of the Korteweg-de Vries Equation for Multiple Collisions of Solitons. Phys. Rev. Lett, 27, 1192.
• Drazin, P. G. , Johnson, R. S. 1989. Solitons: An Introduction. Cambridge University Press. Cambridge.
• Dias, F., Bridges, T., 2004. Weakly nonlinear wave packets and the nonlinear Scrödinger equation. Summer School Graduate Course. Nonlineer waves in fluids. Recent Advancess and Modern Applications, International Centre for Mechanical Sciences. Udine, Italy.
• Zakharov, V. E. Shabat, A. B. 1972. Exact Theory of Two-dimensional Self-focusing and One-dimensional Self-modulation of Wave in Nonlinear Media. Soviet Journal of Experimental and Theoretical Physics,34,62-69.
• Weideman, J. A. C. , Herbst, B. M. 1986. Split-step methods for the solution of the nonlinear Schrödinger equation. SIAM Journal of Numerical Analysis, 23:485–507.
• Overman, E. A. McLaughlin, D. W. Bishop, A. R. 1986. Coherence and chaos in the driven damped Sine-Gordon equation: Measurement of the soliton spectrum. Physica D., 19, 1-41.
• Karaoğlu, B. 2008. Kuantum mekaniğine Giriş. Seçkin Yayıncılık. Ankara.
• Hirota, R. 1973. Exact Envelope-Soliton Solutions of a Nonlinear Wave Equation. Journal of Mathematical Physics, 14(7):805-809.
• Hirota, R. 1972. Exact Solution of the Modified Korteweg-de Vries Equation for Multiple Collisions of Solitons. Journal of the Physical Society of Japan, 33(5):1456-1458.