Double reduction of a nonlinear (2+1) wave equation via conservation laws

Bokhari, Ashfaque H.; Al-Dweik, Ahmad Y.; Kara, A. H.; Mahomed, F. M.; Zaman, F. D.

doi:10.1016/j.cnsns.2010.07.007

Cited by 41 publications

(20 citation statements)

References 7 publications

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“…The conserved vectors obtained here can be used in double reductions and solutions of the underlying equations [20].…”

Section: Discussionmentioning

confidence: 99%

Symmetries and conservation laws of evolution equations via multiplier and nonlocal conservation methods

Yıldırım¹,

Yaşar²

2017

ntmsci

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Abstract:In this work, we have applied a new technique which is a union of multiplier and Ibragimov's nonlocal conservation method for constructing the local conservation laws of nonlinear evolution equations. One can conclude that the higher order solutions of adjoint equation can be obtained by the multiplier functions. The Lax equation and generalized Hirota-Satsuma coupled KdV system are chosen to illustrate the effectiveness of the method. Thus, we have obtained a plenty of local (some of them are the higher order) conservation laws. The combined method presents a wider applicability for handling the conservation laws of nonlinear wave equations.

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“…The conserved vectors obtained here can be used in double reductions and solutions of the underlying equations [20].…”

Section: Discussionmentioning

confidence: 99%

Symmetries and conservation laws of evolution equations via multiplier and nonlocal conservation methods

Yıldırım¹,

Yaşar²

2017

ntmsci

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“…We define a new nonlocal variable w by T t = w x , T x =− w t . Taking the similarity variables T r = w s , T s =− w r , so that the conservation law is written as

D_{r} T^{r} + D_{s} T^{s} = 0,

where ( , )

T^{s} = \frac{T^{t} D_{t} (s) + T^{x} D_{x} (s)}{D_{t} (r) D_{x} (s) - D_{x} (r) D_{t} (s)}

and

T^{r} = \frac{T^{t} D_{t} (r) + T^{x} D_{x} (r)}{D_{t} (r) D_{x} (s) - D_{x} (r) D_{t} (s)} .

The components T x , T t depend upon ( x , t , v , v (1) , v (2) ,…, v ( q −1) ), which means that T s , T r depend upon

(s, r, θ, θ_{r}, θ_{r r}, \dots, θ_{r^{(q - 1)}})

for solutions invariant under X . Therefore, in the new case divergence condition, D r T r + D s T s =0 becomes

\frac{\partial T^{s}}{\partial s} + D_{r} T^{r} = 0

or equivalently

…”

Section: Conservation Lawsmentioning

confidence: 99%

On the exact solutions, lie symmetry analysis, and conservation laws of Schamel–Korteweg–de Vries equation

Giresunlu

Yaşar

2017

Math Methods in App Sciences

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In this work, we study the integrability aspects of the Schamel–Korteweg–de Vries equation that play an important role in studying the effect of electron trapping on the nonlinear interaction of ion‐acoustic waves by including a quasi‐potential. Lie symmetry analysis together with the simplest equation method and Kudryashov method is used to obtain exact traveling wave solutions for this equation. In addition, conservation laws are constructed using two different techniques, namely, the multiplier method and the new conservation theorem. Using the conservation laws and symmetries of the underlying equation, double reduction and exact solution were also constructed. Copyright © 2017 John Wiley & Sons, Ltd.

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“…Sjoberg developed a double reduction theory for a pde of order q with two independent and m dependent variables to reduce it to an ode of order (q-1) provided that the pde admits a conserved vector associated with at least one symmetry [10]. Recently, Bokhari et al [11,12] introduced the generalization of the double reduction theory for pdes, which have independent variables more than two.…”

Section: Introductionmentioning

confidence: 99%

Conservation laws and double reduction of (2+1) dimensional Calogero–Bogoyavlenskii–Schiff equation

San

Akbulut

Ünsal

et al. 2016

Math Methods in App Sciences

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Double reduction of a nonlinear (2+1) wave equation via conservation laws

Cited by 41 publications

References 7 publications

Symmetries and conservation laws of evolution equations via multiplier and nonlocal conservation methods

Symmetries and conservation laws of evolution equations via multiplier and nonlocal conservation methods

On the exact solutions, lie symmetry analysis, and conservation laws of Schamel–Korteweg–de Vries equation

Conservation laws and double reduction of (2+1) dimensional Calogero–Bogoyavlenskii–Schiff equation

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