SOLITON SOLUTIONS OF THE PAINLEVÉ- BÄCKLUND EQUATION USING THE RATIONAL (G'/G) EXPANSION METHOD

Abstract

In this study, the rational (G'/G) expansion method for finding traveling wave solutions of nonlinear formation equations is discussed. Thanks to this method, various soliton solutions in appropriate form arranged according to trigonometric functions, rational functions and hyperbolic functions are obtained. The rational (G'/G) expansion method was applied on the non-linear 1+1-dimensional Painlevé- Bäcklund equation to examine soliton waves that do not disappear when they collide with another wave of the same type. Using this method, traveling wave solutions with arbitrary parameters of the Painlevé-Bäcklund equation are successfully obtained. When special values were given to the parameters, solitary wave solutions of the equations were found from the traveling waves and 3-dimensional and contour graphs were drawn. The proposed rational (G'/G) expansion method is direct, simple and effective. It is an effective and powerful mathematical method for many other nonlinear and integer balancing equations.

Keywords

• Rational Expansion Method
• Painlevé - Bäcklund Equation
• Soliton Solution.

PAINLEVÉ- BÄCKLUND DENKLEMİNİN RASYONEL (G'/G) AÇILIM METODU İLE SOLITON ÇÖZÜMLERİ

Abstract

Bu çalışmada lineer olmayan oluşum denklemlerinin ilerleyen dalga çözümlerinin bulunmasına yönelik rasyonel (G'/G) açılım yöntemi ele alınmıştır. Bu yöntem sayesinde trigonometrik fonksiyonlar, rasyonel fonksiyonlar ve hiperbolik fonksiyonlara göre düzenlenmiş uygun formdaki çeşitli soliton çözümler elde edilir. Aynı türden başka bir dalgayla çarpıştığında yok olmayan soliton dalgalarını incelemek için lineer olmayan 1+1-boyutlu Painlevé- Bäcklund denklemi üzerinde rasyonel (G'/G) açılım yöntemi uygulanmıştır. Bu yöntem kullanılarak Painlevé- Bäcklund denkleminin keyfi parametreleriyle ilerleyen dalga çözümleri başarıyla elde edilir. Parametrelere özel değerler verildiğinde ise ilerleyen dalgalardan denklemlerin soliter dalga çözümleri bulunarak 3-boyutlu ve kontur grafikleri çizdirilmiştir. Önerilen rasyonel (G'/G) açılım yöntemi doğrudan, basit ve etkilidir. Diğer birçok lineer olmayan ve tam sayı dengelenmeye sahip denklemler için etkili ve güçlü bir matematiksel yöntemdir.

Keywords

• Painlevé - Bäcklund Denklemi
• Soliton çözüm
• rasyonel açılım metodu

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