Double reduction of PDEs from the association of symmetries with conservation laws with applications

We use the definition of the association of symmetries of partial differential equations with conservation laws [A. Kara, F. Mahomed, The relationship between symmetries and conservation laws, Int. J. Theor. Phys. 39 (1) (2000) 23–40] to find a double reduction of partial differential equations with two independent variables. Applications to the linear heat equation, the BBM equation, the sine-Gordon equation and a system of differential equations from one dimensional gas dynamics illustrate the new theory.