Parameterized Stationary Solution for first order PDE

Abstract

We analyze the existence of a parameterized stationary solution $z(\lambda ,z_0)=(x(\lambda ,z_0),p(\lambda ,z_0),u(\lambda ,z_0))\in D\subseteq {\mathbb{R}}^{2n+1},\lambda \in B(0,a)\subseteq \prod _{i=1}^m[-a_i,a_i]$, associated with a nonlinear first order PDE, $H_0(x,p(x),u(x))=constant(p(x)={\partial }_{x}u(x))$ relying on (a) first integral $H\in {\mathcal{C}}^{\infty }(B(z_0,2\rho )\subseteq {\mathbb{R}}^{2n+1})$ and the corresponding Lie algebra of characteristic fields is of the finite type; (b) gradient system in a Lie algebra finitely generated over orbits $(f.g.o;z_0)$ starting from $z_0\in D$ and their nonsingular algebraic representation.

9 pages,accepted for publication in Mathematical Reports