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Double reduction of a nonlinear \((2+1)\) wave equation via conservation laws. (English) Zbl 1221.35244
Summary: 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.

MSC:
35L71 Second-order semilinear hyperbolic equations
35A30 Geometric theory, characteristics, transformations in context of PDEs
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