A Kirchhoff equation with infinite conservation laws

TL;DR

This paper demonstrates that the Kirchhoff-Pokhozhaev equation, a quasilinear PDE, possesses conservation laws of all orders, highlighting its rich mathematical structure and invariants.

Contribution

It establishes the existence of infinite conservation laws for the Kirchhoff-Pokhozhaev equation, a novel result in the study of this class of quasilinear PDEs.

Findings

The equation admits conservation laws of all orders.

Infinite conservation laws are established for the Kirchhoff-Pokhozhaev equation.

The result enhances understanding of the equation's mathematical properties.

Abstract

We show here that the quasilinear Kirchhoff-Pokhozhaev equation $$u_{tt}-\big(a\int_{\mathbb{R}^n} |\nabla u |^2 dx + b \big)^{-2} \Delta u = 0,$$ with $a\neq0$, admits conservation laws of all orders.