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 (25)

• Geometry of jet spaces and integrable systems
• A new integrable generalization of the Korteweg - de Vries equation
• Nonholonomic deformation of KdV and mKdV equations and their symmetries, hierarchies and integrability
• Two-fold integrable hierarchy of nonholonomic deformation of the DNLS and the Lenells-Fokas equation
• The bi-Hamiltonian structure and new solutions of KdV6 equation
• Exact accelerating solitons in nonholonomic deformation of the KdV equation with two-fold integrable hierarchy
• Nonlinearizing linear equations to integrable systems including new hierarchies with nonholonomic deformations
• Dynamical study of nonlinear ion-acoustic waves in presence of charged space debris at Low Earth Orbital (LEO) plasma region
• Integrability of Kupershmidt deformations
• Negative Even Grade mKdV Hierarchy and its Soliton Solutions
• Study of quasi-integrable and non-holonomic deformation of equations in the NLS and DNLS hierarchy
• 2+1 KdV(N) Equations
• Hamiltonian structures for general PDEs
• Quasi-periodic solutions to the negative-order KdV hierarchy
• Non-holonomic and Quasi-integrable deformations of the AB Equations
• The generalized Kupershmidt deformation for constructing new integrable systems from integrable bi-Hamiltonian systems
• A connection between HH3 and KdV with one source
• Integrability of one bilinear equation: singularity analysis and dimension
• Study of the family of Nonlinear Schrodinger equations by using the Adler-Kosant-Symes framework and the Tu methodology and their Non-holonomic deformation
• On Negative Order KdV Equations
• Some new integrable systems constructed from the bi-Hamiltonian systems with pure differential Hamiltonian operators
• The generalized Kupershmidt deformation for constructing new discrete integrable systems
• Analysis and comparative study of non-holonomic and quasi-integrable deformations of the Nonlinear Schrödinger Equation
• Study of Non-Holonomic Deformations of Non-local integrable systems belonging to the Nonlinear Schrodinger family
• A study of two new generalized negative KdV type equations