Conservation laws of a nonlinear \((n+1)\) wave equation. (English) Zbl 1197.35174

Summary: Conservation laws of the nonlinear \((n+1)\) wave equation \(u_{tt} = \mathbf{div}(f(u)\mathbf{grad}\,u)\) involving an arbitrary function of the dependent variable, are obtained. This equation is not derivable from a variational principle. By writing the equation, which admits a partial Lagrangian, in the partial Euler-Lagrange form, partial Noether operators associated with the partial Lagrangian are obtained for all possible cases of the arbitrary function. Partial Noether operators are used via a formula in the construction of the conservation laws of the wave equation. If \(f(u)\) is an arbitrary function, we show that there is a finite number of conservation laws for \(n=1\) and an infinite number of conservation laws for \(n\geq 2\). None of the partial Noether operators is a Lie point symmetry of the equation. If \(f\) is constant, where all of the partial Noether operators are point symmetries of the equation, there is also an infinite number of conservation laws.


35L72 Second-order quasilinear hyperbolic equations
35B06 Symmetries, invariants, etc. in context of PDEs
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