Structural and qualitative properties of a geometrically integrable equation

TL;DR

This paper investigates the symmetries, conservation laws, and geometric properties of a specific Novikov integrable equation, providing insights into solution behaviors and associated pseudo-spherical surfaces.

Contribution

It identifies Lie symmetries, conservation laws, and conditions for solution invariance and positivity, and classifies solutions related to pseudo-spherical surfaces, advancing understanding of this integrable equation.

Findings

Lie symmetries of the equation are characterized.

Local conservation laws up to second order are established.

Conditions for $L^1$ norm invariance and positive solutions are provided.

Abstract

Lie symmetries of a Novikov geometrically integrable equation are found and group-invariant solutions are obtained. Local conservation laws up to second order are established as well as their corresponding conserved quantities. Sufficient conditions for the $L^1$ norm of the solutions to be invariant are presented, as well as conditions for the existence of positive solutions. Two demonstrations for unique continuation of solutions are given: one of them is just based on the invariance of the $L^1$ norm of the solutions, whereas the other is based on well-posedness of Cauchy problems. Finally, pseudo-spherical surfaces determined by the solutions of the equation are studied: all invariant solutions that do not lead to pseudo-spherical surfaces are classified and the existence of an analytic metric for a pseudo-spherical surface is proved using conservation of solutions and well-posedness results.