Conserved quantities for (1+2)(1+2)-dimensional non-linear wave equation on curved surfaces

In this paper, relationship between background metric and Noether operators is developed for different surfaces. For this the (1+2)(1+2)-dimensional non-linear wave equation on curved surfaces is considered. The Noether approach is applied on the discussed equation and determining equations for the Noether operators are computed in terms of coefficients of the first fundamental form (FFF). Further these determining equations are utilized to compute the Noether operators and conserved vectors of the considered equation on particular surfaces i.e. sphere (S2)(S2), torus (T2)(T2), flat space (R2)(R2) and cone (C2)(C2). In derivation of conservation laws, two cases of the function f(u)f(u) are observed. For both cases the conserved vectors of the discussed equation on S2S2, T2,R2T2,R2 and C2C2 are established. It is noted that on all discussed surfaces Lie point generators coincide with the corresponding Noether operators while the maximal solvable algebra of symmetries is obtained for f(u)=0f(u)=0.

► A connection between background metric and Noether operators is discussed for different surfaces.
► The Noether approach is applied on the (1 + 2)-dimensional non-linear wave equation on particular surfaces.
► Two cases of the function f(u) are reported and discussed by means of Noether approach.