BEŞİNCİ MERTEBEDEN LİNEER OLMAYAN CAUDREY-DODD-GIBBON DENKLEMİNE ÇOK ÖLÇEKLİ AÇILIM METODU

Yıl 2019, Cilt: 7 Sayı: 1, 59 - 65, 01.01.2019

Murat Koparan

Öz

Lineer olmayan oluşum denklemleri (NLEE) çok sayıda alanda ortaya çıkan problemlerin matematiksel modelleri için temel oluşturur. Geçmiş yıllarda, uygulamalı matematikte oluşum denklemleri önemli bir yer kazanmıştır. Bu çalışma, lineer olmayan oluşum denklemleri (NLEE) için pertürbasyon yöntemi olarak bilinen çok ölçekli açılım metoduyla ilgilidir. Bu raporda, (1 + 1) boyutlu beşinci mertebeden lineer olmayan Caudrey-Dodd-Gibbon (CDG) denkleminin analizi için çok ölçekli açılım metodu uygulanmıştır ve lineer olmayan Schrödinger (NLS) tipi denklemi elde edilmiştir.

Anahtar Kelimeler

Pertürbasyon, Çok ölçekli açılım metodu, Beşinci mertebeden Caudrey-Dodd-Gibbon (CDG) denklemi

A MULTIPLE SCALES METHOD FOR NONLINEAR THE FIFTH-ORDER CAUDREY-DODD-GIBBON EQUATION

Öz

Nonlinear evolution equations form the basis for mathematical models of problems arising in numerous areas. Over the past decades, evolution equations have earned a significant place in applied mathematics. This study relates multiple scale method which is known as a perturbation method for nonlinear evolution equations. In this report, a method of multiple scales is presented for the analysis of the (1+1)-dimensional the fifth-order Caudrey-Dodd-Gibbon (CDG) equation and we derive nonlinear Schrödinger (NLS) type equation.

Anahtar Kelimeler

Perturbation, Multiple scales method, The fifth-order Caudrey-Dodd-Gibbon (CDG) equation

Kaynakça

• [1] Hasegawa A. Plasma Instabilities and Nonlinear Effects. Berlin, Springer-Verlag, 1975.
• [2] Eilenberger G. Solitons. Berlin, Springer-Verlag, 1983.
• [3] Whitham G. Linear and Nonlinear Waves. New York, Wiley, 1974.
• [4] Aiyer RN, Fuchssteiner B, Oevel W. Solitons and Discrete Eigen functions of the Recursion Operator of Nonlinear Evolution Equations: The Caudrey-Dodd- Gibbon-Sawada-Kotera Equations. Journal of Physics A: Mathematical and General 1986; 19: 3755-3770.
• [5] Caudrey PJ, Dodd RK, Gibbon JD. A new heirarchy of Korteweg-de Vries equations. Proc. Roy. Soc. Lond. A 1976; 351: 407- 422.
• [6] Dodd RK, Gibbon JD. The prolongation structure of a higher order Korteweg-de Vries equations. Proc. Roy. Soc. Lond. A (1977; 358: 287-300.
• [7] Zakharov V, Kuznetsov EA. Multiscale expansions in the theory of systems integrable by the inverse scattering transform, Physica D 1986; 18: 455-463.
• [8] Calegora F, Degasperis A, Xiaosdo Ji. Nonlinear Schrödinger-type equations from multiscalereduction of PDEs I. Systematic derivation. J Math Phys 2001; 42: 2635-2652.
• [9] Degasperis A, Manakov SV, Santini PM. Multiple-scale perturbation beyond nonlinear Schrödinger equation I. Physica D 1997; 100: 187-211.
• [10] Osborne AR, Boffetta G. The shallow water NLS equation in Lagrangian coordinates. Phys Fluid A 1989; 1: 1200-1210.
• [11] Osborne AR, Boffetta G. A. Summable multiscale expansion for the KdV equation. In: DegasperisA, Fordy AP, Lakshmanan M, editors. Nonlinear evolution equations. Integrability and spectralMethods. MUP, Manchester and New York, 1991 pp. 559-571.