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

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

doi:10.1016/j.cnsns.2010.07.007

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

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

^*Corresponding author for this work

Research output: Contribution to journal › Article › peer-review

40 Scopus citations

Abstract

Conservation laws of a nonlinear (2+1) wave equation u_tt=(f(u)u_x)_x+ (g(u)u_y)_y involving arbitrary functions of the dependent variable are obtained, by writing the equation in the partial Euler-Lagrange form. Noether-type operators associated with the partial Lagrangian are obtained for all possible cases of the arbitrary functions. If either of f(u) or g(u) is an arbitrary nonconstant function, we show that there are an infinite number of conservation laws. If both f(u) and g(u) are arbitrary nonconstant functions, it is shown that there exist infinite number of conservation laws when f'(u) and g'(u) are linearly dependent, otherwise there are eight conservation laws. Finally, we apply the generalized double reduction theorem to a nonlinear (2+1) wave equation when f'(u) and g'(u) are linearly independent.

Original language	English
Pages (from-to)	1244-1253
Number of pages	10
Journal	Communications in Nonlinear Science and Numerical Simulation
Volume	16
Issue number	3
DOIs	https://doi.org/10.1016/j.cnsns.2010.07.007
State	Published - Mar 2011

Keywords

Conservation laws
Generalized double reduction
Partial Lagrangians
Partial Noether operators

ASJC Scopus subject areas

Numerical Analysis
Modeling and Simulation
Applied Mathematics

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10.1016/j.cnsns.2010.07.007

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title = "Double reduction of a nonlinear (2+1) wave equation via conservation laws",

abstract = "Conservation laws of a nonlinear (2+1) wave equation utt=(f(u)ux)x+ (g(u)uy)y involving arbitrary functions of the dependent variable are obtained, by writing the equation in the partial Euler-Lagrange form. Noether-type operators associated with the partial Lagrangian are obtained for all possible cases of the arbitrary functions. If either of f(u) or g(u) is an arbitrary nonconstant function, we show that there are an infinite number of conservation laws. If both f(u) and g(u) are arbitrary nonconstant functions, it is shown that there exist infinite number of conservation laws when f'(u) and g'(u) are linearly dependent, otherwise there are eight conservation laws. Finally, we apply the generalized double reduction theorem to a nonlinear (2+1) wave equation when f'(u) and g'(u) are linearly independent.",

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author = "Bokhari, \{Ashfaque H.\} and Al-Dweik, \{Ahmad Y.\} and Kara, \{A. H.\} and Mahomed, \{F. M.\} and Zaman, \{F. D.\}",

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doi = "10.1016/j.cnsns.2010.07.007",

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AU - Bokhari, Ashfaque H.

AU - Al-Dweik, Ahmad Y.

AU - Kara, A. H.

AU - Mahomed, F. M.

AU - Zaman, F. D.

PY - 2011/3

Y1 - 2011/3

N2 - Conservation laws of a nonlinear (2+1) wave equation utt=(f(u)ux)x+ (g(u)uy)y involving arbitrary functions of the dependent variable are obtained, by writing the equation in the partial Euler-Lagrange form. Noether-type operators associated with the partial Lagrangian are obtained for all possible cases of the arbitrary functions. If either of f(u) or g(u) is an arbitrary nonconstant function, we show that there are an infinite number of conservation laws. If both f(u) and g(u) are arbitrary nonconstant functions, it is shown that there exist infinite number of conservation laws when f'(u) and g'(u) are linearly dependent, otherwise there are eight conservation laws. Finally, we apply the generalized double reduction theorem to a nonlinear (2+1) wave equation when f'(u) and g'(u) are linearly independent.

AB - Conservation laws of a nonlinear (2+1) wave equation utt=(f(u)ux)x+ (g(u)uy)y involving arbitrary functions of the dependent variable are obtained, by writing the equation in the partial Euler-Lagrange form. Noether-type operators associated with the partial Lagrangian are obtained for all possible cases of the arbitrary functions. If either of f(u) or g(u) is an arbitrary nonconstant function, we show that there are an infinite number of conservation laws. If both f(u) and g(u) are arbitrary nonconstant functions, it is shown that there exist infinite number of conservation laws when f'(u) and g'(u) are linearly dependent, otherwise there are eight conservation laws. Finally, we apply the generalized double reduction theorem to a nonlinear (2+1) wave equation when f'(u) and g'(u) are linearly independent.

KW - Conservation laws

KW - Generalized double reduction

KW - Partial Lagrangians

KW - Partial Noether operators

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ER -

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

Abstract

Keywords

ASJC Scopus subject areas

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