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: u(tt) - uu(xx) + 3/4(u(x))(2) + u(2) + ($) over bar au(xx) + ($) over bar bu(xxxx) = 0, and quasilinear heat equations: v(t) = (v(-4/3)v(x))(x) + ($) over bar av(-1/3) + ($) over bar bv(7/3) + ($) over bar cv, v(t) = del . (v(sigma)($) over bar Vv) + ($) over bar av(1-sigma) + ($) over bar bv. 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.