Abstract.
We lay down the foundations of the theory of Poisson vertex algebras aimed at its applications to integrability of Hamiltonian partial differential equations. Such an equation is called integrable if it can be included in an infinite hierarchy of compatible Hamiltonian equations, which admit an infinite sequence of linearly independent integrals of motion in involution. The construction of a hierarchy and its integrals of motion is achieved by making use of the so called Lenard scheme. We find simple conditions which guarantee that the scheme produces an infinite sequence of closed 1-forms \(\omega_j, j \in {\mathbb {Z}}_+\), of the variational complex Ω. If these forms are exact, i.e., ω j are variational derivatives of some local functionals ∫ h j , then the latter are integrals of motion in involution of the hierarchy formed by the corresponding Hamiltonian vector fields. We show that the complex Ω is exact, provided that the algebra of functions is \(\fancyscript {V}\) is “normal”; in particular, for arbitrary \(\fancyscript {V}\), any closed form in Ω becomes exact if we add to \(\fancyscript {V}\) a finite number of antiderivatives. We demonstrate on the examples of the KdV, HD and CNW hierarchies how the Lenard scheme works. We also discover a new integrable hierarchy, which we call the CNW hierarchy of HD type. Developing the ideas of Dorfman, we extend the Lenard scheme to arbitrary Dirac structures, and demonstrate its applicability on the examples of the NLS, pKdV and KN hierarchies.
Similar content being viewed by others
References
Bakalov B., Kac V.G. and Voronov A.A. (1999). Cohomology of conformal algebras. Comm. Math. Phys. 200: 561–598
Barakat A. (2008). On the moduli space of deformations of bihamiltonian hierarchies of hydrodynamic type. Adv. Math. 219: 604–632
Sole De A. and Kac V.G. (2006). Finite vs affine W-algebras, Jpn. J. Math. 1: 137–261
A. De Sole and V.G. Kac, Lie conformal algebra cohomology and the variational complex, Comm. Math. Phys., 292 (2009), 667–719; arXiv:0812.4897.
L.A. Dickey, Soliton Equations and Hamiltonian Systems. Second ed., Adv. Ser. Math. Phys., 26, World Sci. Publ., 2003.
I. Dorfman, Dirac Structures and Integrability of Nonlinear Evolution Equations, Nonlinear Sci. Theory Appl., John Wiley & Sons, 1993.
Gelfand I.M. and Dorfman I. (1980). Schouten bracket and Hamiltonian operators. Funktsional Anal. i Prilozhen. 14: 71–74
Gelfand I.M. and Dorfman I. (1982). Hamiltonian operators and the classical Yang–Baxter equation. Funktsional Anal. i Prilozhen. 16: 1–9
Helmholtz H. (1887). Über die physikalische Bedeutung des Prinzips der Kleinsten Wirkung. J. Reine Angew. Math., 100: 137–166
Ito M. (1982). Symmetries and conservation laws of a coupled nonlinear wave equation. Phys. Lett. A, 91: 335–338
V.G. Kac, Vertex Algebras for Beginners, Univ. Lecture Ser., 10, Amer. Math. Soc., Providence, RI, 1996; Second ed., 1998.
Kosmann-Schwarzbach Y. (1996). From Poisson algebras to Gerstenhaber algebras. Ann. Inst. Fourier (Grenoble) 46: 1243–1274
Magri (1978). A simple model of the integrable Hamiltonian equation. J. Math. Phys. 19: 1156–1162
P. Olver, Applications of Lie Groups to Differential Equations, Grad. Texts in Math., 107, Springer-Verlag, 1986.
Sokolov V.V. (1984). On Hamiltonian property of the Krichever–Novikov equation, Dokl. Akad. Nauk SSSR, 277: 48–50
L.A. Takhtazhyan and L.D. Faddeev, Hamiltonian Approach in Soliton Theory, Nauka, 1986.
Vinogradov A.M. (1977). On the algebra-geometric foundations of Lagrangian field theory. Sov. Math. Dokl., 18: 1200–1204
V. Volterra, Leçons sur les Fonctions de Lignes, Gauthier-Villar, Paris, 1913.
Wilson G. (1988). On the quasi-Hamiltonian formalism of the KdV equation. Phys. Lett. A, 132: 445–450
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by: Yasuyuki Kawahigashi
About this article
Cite this article
Barakat, A., De Sole, A. & Kac, V.G. Poisson vertex algebras in the theory of Hamiltonian equations. Jpn. J. Math. 4, 141–252 (2009). https://doi.org/10.1007/s11537-009-0932-y
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11537-009-0932-y
Keywords and phrases:
- evolution equation
- evolutionary vector field
- local functional
- integral of motion
- integrable hierarchy
- normal algebra of differential functions
- Lie conformal algebra
- Poisson vertex algebra
- compatible λ -brackets
- Lenard scheme
- Beltrami λ -bracket
- variational derivative
- Fréchet derivative
- variational complex
- Dirac structure
- compatible Dirac structures