ABSTRACT
Recall that, given an operator F , a set M ⊆ Wn is said to be invariant on the linear subspace Wn , and then Wn is partially invariant if
F[M] ⊆ Wn .
7.1 Partial invariance for polynomial operators
7.1.1 Basic ideas and examples
Let us begin with an extension of the results on invariant subspaces in Section 1.5.2, where we considered general quadratic operators
F[u] =∑(i, j ) ai, j Dix u D jx u, (7.1) with the real symmetric matrix ‖ai, j‖ and the corresponding polynomial
P(X,Y ) =∑(i, j ) ai, j X i Y j . We are looking for ODE reductions of PDEs with the operator (7.1) on the subspace
Wn = L{epk x , k = 1, ..., n}, where, for convenience, p1 = 0. Let = {p1, ..., pn} denote the set of all the exponents. Such finite-dimensional subspaces are natural for N-soliton solutions of many integrable PDEs, including the KdV, Harry Dym, and Boussinesq equations (see Section 4.1). Here we construct exact solutions of non-integrable PDEs.
