2007/09/28 by Boris A. Kupershmidt · 1 citation
Physics and Astronomy · #nlin.SI
paper · pdf · doi:10.1016/j.physleta.2007.12.019
arxiv created 2007/09/28 · arxiv updated 2009/12/01
K2 S2 T [5] recently derived a new 6th-order wave equation KdV6: (∂2x + 8ux ∂x + 4uxx)(ut + uxxx + 6ux2) = 0, found a linear problem and an auto-B\rmackclund transformation for it, and conjectured its integrability in the usual sense. We prove this conjecture by constructing an infinite commuting hierarchy KdVn6 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.