KdV6: An Integrable System

Abstract

$K^2{S}^{2}T[5]$ recently derived a new 6${}^{th}$-order wave equation $KdV6$: $({\partial }_x^2+8u_x{\partial }_x+4u_{xx})(u_t+u_{xxx}+6u_x^2)=0$, found a linear problem and an auto-B$\ddot{\mathrm{a}}$ckclund transformation for it, and conjectured its integrability in the usual sense. We prove this conjecture by constructing an infinite commuting hierarchy $Kd{V}_{n}6$ 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.

References in corpus (1)

• On the History of the Development of the Nonholonomic Dynamics

Cited by in corpus (3)

• Nonholonomic deformation of KdV and mKdV equations and their symmetries, hierarchies and integrability
• The bi-Hamiltonian structure and new solutions of KdV6 equation
• Hamiltonian structures for general PDEs