KdV6: An Integrable System

TL;DR

This paper proves the integrability of the recently derived KdV6 equation by constructing an infinite hierarchy and conserved densities, and introduces a general method for nonholonomic perturbations of bi-Hamiltonian systems.

Contribution

It establishes the integrability of KdV6 and presents a general construction for nonholonomic perturbations applicable to bi-Hamiltonian systems.

Findings

Proved KdV6's integrability via infinite hierarchy and conserved densities.

Developed a method for nonholonomic perturbations of bi-Hamiltonian systems.

Verified the conjecture in classical long-wave, Toda lattice, and Euler top cases.

Abstract

$K^2 S^2 T [5]$ recently derived a new 6$^{th}$-order wave equation $KdV6$: $(\partial^2_x + 8u_x \partial_x + 4u_{xx})(u_t + u_{xxx} + 6u_x^2) = 0$, found a linear problem and an auto-B${\ddot{\rm{a}}}$ckclund transformation for it, and conjectured its integrability in the usual sense. We prove this conjecture by constructing an infinite commuting hierarchy $KdV_n6$ with a common infinite set of conserved densities. A general construction is presented applicable to any bi-Hamiltonian system (such as all standard Lax equations, continuous and discrete) providing a nonholonomic perturbation of it. This perturbation is conjectured to preserve integrability. That conjecture is verified in a few representative cases: the classical long-wave equations, the Toda lattice (both continuous and discrete), and the Euler top.