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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 Semilinear second-order hyperbolic equations 35A30 Geometric theory for PDE, characteristics, transformations
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##### References:
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