[Conserved quantities in] PDEs

In the early 1960s it was discovered that certain nonlinear PDEs support an infinite number of distinct conserved quantities, associated with so-called integrability and the presence of solitons. Systematic methods now exist to find conserved quantities that are given by integrals of polynomials of any given degree in the dependent variables and their derivatives. Most randomly chosen PDEs appear, however, to have no such conserved quantities.