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

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Abstract
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.