Invariant subspaces and new explicit solutions to evolution equations with quadratic nonlinearities

We present new explicit solutions to some classes of quasilinear evolution equations arising in different applications, including equations of the Boussinesq type: and quasilinear heat equations: The method is based on construction of finite-dimensional linear functional subspaces which are invariant with respect to spatial operators having quadratic nonlinearities. The corresponding nonlinear evolution equations on invariant subspaces are shown to be equivalent to finite-dimensional dynamical systems. Examples of two-, three- and five- dimensional invariant subspaces are given. Some generalisations to N -dimensional quadratic operators are also considered.