Second-order nonlinear differential operators possessing invariant subspaces of submaximal dimension

The invariant subspace method is used to construct the explicit solution of a nonlinear evolution equation. The second-order nonlinear differential operators that possess invariant subspaces of submaximal dimension are described. There are second-order nonlinear differential operators, including cubic operators and quadratic operators, which preserve an invariant subspace of submaximal dimension. A full description of the second-order cubic operators with constant coefficients admitting a four-dimensional invariant subspace is given. It is shown that the maximal dimension of invariant subspaces preserved by a second-order cubic operator is four. Several examples are given for the construction of the exact solutions to nonlinear evolution equations with cubic nonlinearities. These solutions blow up in a finite time.