ABSTRACT

We say that the first-order operator H[u] = H (x, u,∇u, ut) is a sign-invariant (SI) of the equation (8.1) if, for any solution u(x, t), the following holds:

(1) H[u(x, 0)] ≥ 0 in IRN ⇒ H[u(x, t)] ≥ 0 in IRN for t > 0, (2) H[u(x, 0)] ≤ 0 in IRN ⇒ H[u(x, t)] ≤ 0 in IRN for t > 0,

i.e., both signs of H[u] are preserved in evolution. Such an evolution invariance of signs is controlled by the Maximum Principle (MP). Since, by definition, any signinvariantH[u] is also the zero-invariant, i.e.,

H[u(x, 0)] = 0 in IRN ⇒ H[u(x, t)] = 0 in IRN for t > 0, (8.2) this makes it possible to construct exact solutions of equations (8.1) if we know how to integrate the first-order PDE in (8.2). We will derive finite and, in some cases, infinite-dimensional sets of equations (8.1) possessing solutions that are expressed in terms of dynamical systems or algebraic relations. It turns out that these solutions often belong to some linear invariant subspaces or sets.