On the conservation laws for a certain class of nonlinear wave equation via a new conservation theorem

Bokhari et al. [14] applied the partial Noether approach to find the Noether-type operators and conservation laws for a certain class of the nonlinear (2 + 1) wave equation utt = (f(u)ux)x + (g(u)uy)y describing waves in two dimension involving arbitrary velocity functions. These arbitrary functions generally arise when transmitting a signal on a transmission plane with material properties that are changing along the plane. Recently, Ibragimov in [15] presented nonlocal conservation theorem method. In this study, we investigate conservation laws via the new conservation theorem. We show that Y = β ( x ) ∂ ∂ v is a Noether symmetry for the system consisting of the PDE and the its adjoint equation with respect to the formal Lagrangian, where v = β(x) is a solution for the adjoint equation. Then we show that the basic set of conservation laws that is estimated by the partial Noether approach can be given by the new conservation theorem if we consider the Noether symmetry β ( x ) ∂ ∂ v .We note that the classification of the variable v in the adjoint equation in the new conservation theorem approach is equivalent to the classification of the characteristics of the Noether type operator in the partial Noether approach. New four sets of conservation laws corresponding to translations and scaling symmetries are constructed that did not appear before. We show that each one of these new four sets is linearly independent set and form a subgroup of conservation laws.