Parameterized Stationary Solution for first order PDE

TL;DR

This paper investigates the existence of parameterized stationary solutions for a class of nonlinear first order PDEs, utilizing Lie algebra structures and first integrals to establish conditions for solutions.

Contribution

It introduces a novel approach using Lie algebra of characteristic fields and first integrals to analyze stationary solutions of first order PDEs.

Findings

Existence of parameterized stationary solutions under Lie algebra conditions

Use of first integrals and gradient systems to characterize solutions

Application of algebraic and geometric methods to PDE analysis

Abstract

We analyze the existence of a parameterized stationary solution $z(\lambda,z_0)=\big(x(\lambda,z_0), p(\lambda,z_0),\,u(\lambda,z_0)\big)\in D\subseteq\mathbb{R}^{2n+1},\,\lambda\in B(0,a)\subseteq\mathop{\prod}\limits_{i=1}^{m}[-a_i,a_i]$, associated with a nonlinear first order PDE, $H_0(x,p(x),u(x))=\hbox{constant}\,\,(p(x)=\partial_x u(x))$ relying on (a) first integral $H\in\mathcal{C}^\infty\big(B(z_0,2\rho)\subseteq\mathbb{R}^{2n+1}\big)$ 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.