Üçüncü Mertebe Dispersiyon Terimli Beşli-Yedili Doğrusal Olmayan Schrödinger Denklemi

Bu çalışmada, genelleştirilmiş, kübik doğrusal olmayan elastik bir ortamda yayılan boyuna bir dalgada, yüksek mertebeden dağılım ve doğrusal olmayan etkilerin katkılarını incelemek için beşli-yedili doğrusal olmayan modülasyonu düşünülmektedir. Son zamandaki çalışmalarda, hemen hemen tek dalga sayılı dalga çözümlerinin çok ölçekli açılımı kullanılarak boyuna bir dalganın modülasyonu için üçüncü mertebeden dispersiyon terimli kübik, doğrusal olmayan Schrödinger denklemi elde edildi. Elde edilen denklemde bazı doğrusal olmayan terimlerin katsayılarının yer almadığı belirli bir malzeme sabiti ve dalga sayısı değerleri seçilirse, boyuna bir dalganın davranışını tanımlamak için, doğrusal olmayan etkileri dağılım etkilerle dengelendiği yeni bir pertürbasyon açılımına ihtiyaç vardır. Sonuç olarak, üçüncü dereceden dağılım terimli beşli-yedili doğrusal olmayan Schrödinger denklemi, beşli-yedili doğrusal olmayan etkinin üçüncü mertebeden dağılım terimiyle dengelendiği yeni bir model olarak elde edilir.

Anahtar Kelimeler:

Doğrusal Olmayan Schrödinger Denklemleri, Doğrusal Olmayan Dalga Yayılımı, Genelleştirilmiş Elastik Ortam

Quintic-Septic Nonlinear Schrödinger Equation with a Third-Order Dispersion Term

In the present study, the quintic-septic nonlinear modulation of a longitudinal wave propagating to contribute the dispersive and higher-order nonlinear effects in a generalized cubically nonlinear elastic medium is considered. In recent work, for the modulation of a longitudinal wave, a cubic nonlinear Schrödinger equation with a third-order dispersive term is obtained by using a multi-scale expansion of quasi-monochromatic wave solutions. The third- quintic-septic longitudinal wave, by choosing specific values of material constants and wave number for which some coefficients of nonlinear terms are disappeared. In this case, a new perturbation expansion is needed to balance nonlinear effects with dispersive effects. As a result, a quintic-septic nonlinear Schrödinger equation with a third-order dispersion term is obtained as a new model that balances quintic-septic nonlinearity with a third-order dispersion term.

Keywords:

: Nonlinear Schrödinger Equations, Nonlinear Wave Propagation, Generalized Elastic Medium,

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M. J. Ablowitz, Nonlinear Dispersive Waves: Asymptotic Analysis and Solitons. Cambridge, New York: Cambridge University Press, 2011.
G. P. Agrawal, Nonlinear Fiber Optics. Oxford: Academic Press, 2013.
V. I. Erofeyev and A. I. Potapov, “Nonlinear wave processes in elastic media with inner structure,” in IV International Workshop on Nonlinear and Turbulent Processes in Physics, Kiev, 1989, pp. 1196–1215.
C. Sulem and P. L. Sulem, The Nonlinear Schrödinger Equation: Self-Focusing and Wave Collapse. New York: Springer-Verlag, 1999.
G. Fibich, The Nonlinear Schrödinger Equation: Singular Solutions and Optical Collapse. Switzerland: Springer International Publishing, 2015.
I. Hacinliyan, “A higher-order long longitudinal wave and short longitudinal wave equations in a generalized elastic medium,” OHU J. Eng. Sci., 8, 138-148, 2019.
S. Erbay, “Modulation of waves near the marginal state of instability in fluid-filled distensible tubes,” J. Phys. A: Math. Gen., 28, 2905-2919, 1995.
B. G. Onana Essama, J. Atangana, B. Mokhtari, N. Cherkaoui Eddeqaqi, and T. C. Kofane, “Theoretical model for electromagnetic wave propagation in negative-index material induced by cubic-quintic nonlinearities and third-order dispersion effects,” Opt. Quant. Electron, 46, 911-924, 2014.
M. Kerbouche, Y. Hamaizi, A. El-Akrmi, and H. Triki, “Solitary wave solutions of the cubic-quintic-septic nonlinear Schrödinger equation in fiber Bragg gratings,” Optik, 127, 9562-9570, 2016.
Q. Zhou, D-Z Yao and Z. Cui, “Exact solutions of the cubic-quintic nonlinear optical transmission equation with higher-order dispersion terms and self-steepening term,” J. Mod. Opt., 59, 57-60, 2011.
Q. Zhou and Q. Zhu, “Optical solitons in medium with parabolic law nonlinearity and higher-order dispersion,” Wave Random Complex, 25, 52-59, 2015.
M. Saha and A. K. Sarma, “Solitary wave solutions and modulation instability analysis of the nonlinear Schrödinger equation with higher-order dispersion and nonlinear terms,” Commun. Nonlinear Sci. Numer. Simulat., 18, 2420-2425, 2013.
F. Azzouzi, H. Triki, and Ph. Grelu, “Dipole soliton solution for the homogeneous high-order nonlinear Schrödinger equation with cubic–quintic–septic non-Kerr terms,” Appl. Math. Model, 39, 1300-1307, 2015.
T. Taniuti, “Reductive perturbation methods and far-fields of wave equation,” Prog. Theor. Phys. Suppl., 55, 1-35, 1974.
A. Jeffrey, and T. Kawahara, Asymptotic Methods in Nonlinear Wave Theory, Boston Mass: Pitman Advanced Publishing Program, 1982.
L. Ostrovsky and K. Gorshkov, “Perturbation theories for nonlinear waves (Book style with paper title and editor),” in Nonlinear Science at the Dawn of the 21st Century, P. L. Christiansen, M. P. Sorensen, A. C. Scott, Eds. Berlin Heidelberg: Springer-Verlag, 2000, pp. 47–66.
L. Ostrovsky, Asymptotic Perturbation Theory of Waves, Singapore: Imperial College Press, 2015.