Invariant reduction for PDEs. I: Conservation laws of 1+1 systems of evolution equations

Among various methods for constructing exact solutions of partial differential equations, the symmetry-invariant approach is particularly noteworthy. This method is especially effective in the case of point symmetries, but when it comes to higher symmetries, additional steps are required to obtain invariant solutions. It turns out that systems that describe symmetry-invariant solutions inherit symmetry-invariant geometric structures even in the case of higher symmetries. Moreover, the reduction of invariant conservation laws of 1+1 systems of evolution equations can be described as an algorithm and implemented in Maple. Starting from invariant conservation laws, we get constants of invariant motion. They are analogs of first integrals of ODEs, and one can use them in the same way. In particular, a sufficient number of independent constants of invariant motion allows one to integrate the corresponding system for invariant solutions.