分野別セミナー

Generally speaking, classical Lie groups of symmetries leave invariant the solution set of a target partial differential equation. However, to find an invariant solution, one needs only find a solution of the PDE that is compatible with a symmetry invariant surface condition. Such a symmetry need not leave the whole solution set invariant. These non-classical symmetries lead to a set of nonlinear determining relations, whose solutions may include more than the solution set of Lie’s classical linear determining relations. By using this approach, we have constructed the first exact solutions of the multi-dimensional nonlinear Richards equation with a plant root extraction term. Examples are given for irrigation through periodic furrows and for water supply through a rotating circular sprinkler.