[Paper Review] A fifth-order bi-Hamiltonian system

Daryoush Talati|arXiv (Cornell University)|Apr 7, 2013

Nonlinear Waves and Solitons1 references3 citations

TL;DR

This paper introduces a new two-component fifth-order bi-Hamiltonian system that reduces to the Kupershmidt equation when $ v = 0 $, establishing its integrability via a compatible pair of Hamiltonian operators $ J $ and $ K $. The system is shown to support an infinite hierarchy of symmetries and conservation laws through the Magri scheme, with explicit expressions for conserved densities and recursion operators, marking the first such system with this reduction property.

ABSTRACT

In this work, we introduce a new two component fifth-order bi-Hamiltonian sys- tem admitting the scalar Kupershmidt equation as a reduction.

Motivation & Objective

• To construct a new two-component fifth-order evolution system that reduces to the Kupershmidt equation when $ v = 0 $, a case previously lacking known integrable examples.
• To establish the bi-Hamiltonian structure of the system using a compatible pair of Hamiltonian operators $ J $ and $ K $, ensuring integrability via the Magri scheme.
• To derive explicit expressions for the first few conserved densities $ \rho_{-1}, \rho_0, \rho_1, \rho_2 $, confirming the existence of an infinite hierarchy of symmetries.
• To demonstrate that the recursion operator $ R = KJ^{-1} $ generates higher symmetries from basic ones, validating the system's complete integrability.

Proposed method

• The system is derived by constructing a fifth-order two-component evolution equation with specific nonlinear terms in $ u $, $ v $, and their derivatives, ensuring reduction to the Kupershmidt equation upon setting $ v = 0 $.
• The bi-Hamiltonian structure is established using two compatible Hamiltonian operators: $ J = \text{diag}(3D_x, D_x) $ and a more complex $ K $-operator with differential and nonlocal terms involving $ D_x^{-1} $.
• The operators $ J $ and $ K $ are verified to commute via the vanishing of the functional trivector of $ K + \lambda J $, confirming compatibility.
• Conserved densities $ \rho_n $ are computed explicitly up to $ \rho_2 $, with $ \rho_{-1} = \alpha $, $ \rho_0 = u^2 + 3v^2 $, and higher ones involving high-order derivatives and nonlinear combinations.
• The recursion relation $ F_n = J \delta \rho_n / \delta (u,v) = K \delta \rho_{n-2} / \delta (u,v) $ is used to generate an infinite hierarchy of symmetries.
• The recursion operator $ R = KJ^{-1} $ is applied iteratively to the basic symmetries $ (u_x, v_x) $ and $ (u_t, v_t) $, producing generalized symmetries at each step.

Experimental results

Research questions

• RQ1 Can a two-component fifth-order bi-Hamiltonian system be constructed that reduces to the Kupershmidt equation when $ v = 0 $?
• RQ2 What is the explicit form of the compatible Hamiltonian pair $ (J, K) $ that realizes the Magri scheme for this system?
• RQ3 What are the conserved densities $ \rho_n $ that generate the infinite hierarchy of symmetries?
• RQ4 How can the recursion operator $ R = KJ^{-1} $ be used to generate higher symmetries from lower ones?
• RQ5 Is the functional trivector of $ K + \lambda J $ vanishing for all $ \lambda $, confirming the compatibility of $ J $ and $ K $?

Key findings

• The system (4) reduces to the Kupershmidt equation $ u_t = u_{5x} + 5u_x u_{3x} + 5u_{xx}^2 - 5u^2 u_{3x} - 20u u_x u_{xx} - 5u_x^3 + 5u^4 u_x $ when $ v = 0 $, confirming the desired reduction.
• The Hamiltonian operators $ J = \text{diag}(3D_x, D_x) $ and $ K $ are compatible, as verified by the vanishing of the functional trivector of $ K + \lambda J $ for all $ \lambda $, a necessary condition for bi-Hamiltonian integrability.
• The first few conserved densities are explicitly computed: $ \rho_{-1} = \alpha $, $ \rho_0 = u^2 + 3v^2 $, and $ \rho_1 $, $ \rho_2 $ involve high-order derivatives and nonlinear terms up to eighth order in $ u $ and $ v $.
• The recursion operator $ R = KJ^{-1} $ generates generalized symmetries from the basic ones $ (u_x, v_x) $ and $ (u_t, v_t) $, confirming the existence of an infinite hierarchy of symmetries.
• The Magri scheme $ F_n = J \delta \rho_n / \delta (u,v) = K \delta \rho_{n-2} / \delta (u,v) $ holds for the system, proving integrability via the Lenard-Magri chain.
• The system is the first known two-component fifth-order bi-Hamiltonian system with reduction to the Kupershmidt equation, filling a gap in the classification of such systems.