## 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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### References:

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