## 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.

### MSC:

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

### References:

 [1] Ovsiannikov, L.V., Group analysis of differential equations, (1982), Academic Press New York · Zbl 0485.58002 [2] Ibragimov, N.H., Transformation groups applied to mathematical physics, (1983), Nauka Moscow, English translation by D. Reidel, Dordrecht (1985) [3] Bluman, G.W.; Kumei, S., Symmetries and differential equations, (1989), Springer-Verlag New York · Zbl 0698.35001 [4] Olver, P.J., Application of Lie groups to differential equations, (1993), Springer-Verlag New York [5] Ibragimov, N.H., () [6] Ibragimov, N.H.; Kara, A.H.; Mahomed, F.M., Lie-Bäcklund and Noether symmetries with applications, Nonlinear dynam., 15, 115-136, (1998) · Zbl 0912.35011 [7] Kara, A.H.; Mahomed, F.M., Relationship between symmetries and conservation laws, Int. J. theor. phys., 39, 23-40, (2000) · Zbl 0962.35009 [8] Noether, E., Invariante variationsprobleme, Nachr. König. gesell. wissen., Göttingen, math. phys. kl. heft, 2, 235-257, (1918), English translation in Transport Theory Statist. Phys. 1(3), 186-207 (1971) · JFM 46.0770.01 [9] Bessel-Hagen, E., Über die erhaltungssätze der elektrodynamik, Math. ann., 84, 258-276, (1921) · JFM 48.0877.02 [10] Kara, A.H.; Mahomed, F.M., Noether-type symmetries and conservation laws via partial Lagrangians, Nonlinear dynam., 45, 367-383, (2006) · Zbl 1121.70014 [11] Khalique, C.M.; Mahomed, F.M., Conservation laws for equations related to soil water equations, Math. probl. engin., 26, 1, 141-150, (2005) · Zbl 1079.35004 [12] Anco, S.C.; Bluman, G.W., Direct construction method for conservation laws of partial differential equations, part II: general treatment, Eur. J. appl. math., 9, 567-585, (2002) · Zbl 1034.35071 [13] Ibragimov, N.H.; Kolsrud, T., Lagrangian approach to evolution equations: symmetries and conservation laws, Nonlinear dynam., 36, 1, 29-40, (2004) · Zbl 1106.70012 [14] Kara, A.H.; Mahomed, F.M., A basis of conservation laws for partial differential equations, J. nonlinear math. phys., 9, 60-72, (2002) · Zbl 1362.35024 [15] A.G. Johnpillai, A.H. Kara, F.M. Mahomed, Conservation laws of a nonlinear $$(1 + 1)$$ wave equation. Nonlinear Anal. B., in press (doi:10.1016/j.nonrwa.2009.06.013) · Zbl 1194.35272
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.