Double reduction of a nonlinear (2+1) wave equation via conservation laws |
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| Institution: | 1. Department of Basic Science, Faculty of Engineering at Benha, Benha University, Benha, Egypt;2. Department of Mathematics, Zhejiang Normal University, Jinhua 321004, Zhejiang, China;3. Department of Mathematics, King Abdulaziz University, Jeddah 21589, Saudi Arabia;4. Department of Mathematics and Statistics, University of South Florida, Tampa, FL 33620-5700, USA;5. School of Mathematics, South China University of Technology, Guangzhou 510640, China;6. Department of Mathematics, Zagazig Faculty of Engineering, Zagazig University, Zagazig, Egypt |
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| 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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