Exact and numerical solutions of generalized Drinfeld–Sokolov equations

We study the bifurcations of nonlinear waves for the generalized Drinfel’d-Sokolov systemut+(vm)x=0,vt+a(vn)xxx+buxv+cuvx=0calledD(m,n)system. We reveal some interesting bifurcation phenomena as follows. (1) ForD(2,1)system, the fractional solitary waves can be bifurcated from the trigonometric periodic waves and the elliptic periodic waves, and the kink waves can be bifurcated from the solitary waves and the singular waves. (2) ForD(1,2)system, the compactons can be bifurcated from the solitary waves, and the peakons can be bifurcated from the solitary waves and the singular cusp waves. (3) ForD(2,2)system, the solitary waves can be bifurcated from the smooth periodic waves and the singular periodic waves.

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“…Completing the integrals above and solving the equations for , we obtain the solutions V 1 ( ), 1 ( ) (see (22)) and V 2 ( ), 2 ( ) (see (24)).…”

Section: Propositionmentioning

confidence: 99%

“…is considered in [14][15][16][17][18][19][20][21][22]. In [23][24][25][26][27][28][29][30][31], many exact solutions of system (7) were obtained. Clearly, system (1) and system (7) are two different systems.…”

Section: Introductionmentioning

confidence: 99%

Some Interesting Bifurcations of Nonlinear Waves for the Generalized Drinfel’d-Sokolov System

Cai

Pan

Liu

2014

Abstract and Applied Analysis

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“…However, what the key of problem is, the different type of nonlinear differential equation can only be efficiently solved by different corresponding methods. For instant, the Jacobi elliptic function expansion method [7], the exp-function method [8], the F-expansion method [9], the tanh method [10], the auxiliary equation method [11], the simplest equation method [12], and linearized harmonic balance method [13] are reliable methods for obtaining exact solutions of specific types of nonlinear differential equations.…”

Section: Introductionmentioning

confidence: 99%

A New Method for the Exact Solution of Duffing Equation

Nelson¹,

Zheng²,

Dorian³

et al. 2018

JAMP

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A lot of methods, such as Jacobian elliptic function analysis, are used to look for the explicit exact solution of Duffing differential equation. The key of the analysis is to construct quotient trigonometric function, and then nonlinear algebraic equation set theory and method are used for the solution of some kinds of nonlinear Duffing differential equation. In this paper, the exact solution of Duffing equation is obtained by using constant variation method, making use of the formula to solve cubic equations and general solution of the homogeneous equation of Duffing equation with appropriate Constant m and function () f t .

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“…Denklemin fiziksel uygulaması ayrıntılı olarak Kaynak [2]'de ele alınmıştır. DS denklem sistemi ve çözümleri çeşitli araştırmacılar tarafından farklı yöntemler kullanılarak incelenmiştir [3][4][5][6][7][8]. DS denklem sistemi; ve bağımlı ve ve bağımsız değişkenler, , ve keyfi sabitler olmak üzere + ( 2 ) = 0…”

Section: Introductionunclassified

Li̇neer Olmayan Kismi̇ Türevli̇ Denklemlere Homotopi̇ Pertürbasyon Yöntemi̇ni̇n Uygulamasi

Dağhan¹,

Yıldız²

2016

Ömer Halisdemir Üniversitesi Mühendislik Bilimleri Dergisi

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Bu çalışmada, lineer olmayan Drinfeld-Sokolov denklem sistemi ve Modifiye-Benjamin-Bona-Mahony denkleminin pertürbatif çözümleri, homotopi pertürbasyon yöntemi kullanılarak elde edilmiştir. Denklemlerin pertürbatif çözümleri için üç iterasyon yapılmıştır. Birinci iterasyonlar kullanılarak her iki denklem için de basit dallanma noktaları hesaplanmıştır. İkinci iterasyonlar için basit dallanma noktasının yalnızca Drinfeld-Sokolov denklem sisteminde olduğu görülmüştür.

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Exact and numerical solutions of generalized Drinfeld–Sokolov equations

Cited by 21 publications

References 35 publications

Some Interesting Bifurcations of Nonlinear Waves for the Generalized Drinfel’d-Sokolov System

Some Interesting Bifurcations of Nonlinear Waves for the Generalized Drinfel’d-Sokolov System

A New Method for the Exact Solution of Duffing Equation

Li̇neer Olmayan Kismi̇ Türevli̇ Denklemlere Homotopi̇ Pertürbasyon Yöntemi̇ni̇n Uygulamasi

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