Abstract
The multisymplectic geometry for the Zakharov–Kuznetsov equation is presented in this Letter. The multisymplectic form and the local energy and momentum conservation laws are derived directly from the variational principle. Based on the multisymplectic Hamiltonian formulation, we derive a 36-point multisymplectic integrator.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Betounes, D. E.: Extension of the classical Cartan form, Phys. Rev. D 29 (1984), 599–606.
Bridges, T. J.: Multi-symplectic structures and wave propagation, Math. Proc. Cambridge Philos. Soc. 121 (1997), 147–190.
Bridges, T. J.: A geometric formulation of the conservation of wave action and its implications for signature and the classification of instabilities, Proc. Royal. Soc. London A 453 (1997), 1365–1395.
Bridges, T. J.: Universal geometric condition for the transverse instability of solitary waves, Phys. Rev. Lett. 84 (2000), 2614–2617.
Bridges, T. J. and Reich, S.: Multi-symplectic integrators: numerical schemes for Hamiltonian PDEs that conserve symplecticity, Phys. Lett. A 284 (2001), 184–193.
Bridges, T. J. and Reich, S.: Multi-symplectic spectral discretizations for the Zakharov–Kuznetsov and shallow water equations, Physica D 152 (2001), 491–504.
Chen, J. B.: Total variation in discrete multisymplectic field theory and multisymplecticenergy-momentum integrators, Lett. Math. Phys. 61 (2002), 63–73.
Infeld, E., Skorupski, A. A. and Senatorski, A.: Dynamics of waves and multidimensional solitons of the Zakharov–Kuznetsov equation, J. Plasma Phys. 64 (2000), 397–409.
Iwasaki, S., Toh, and Kawahara, T.: Cylindrical qusi-solitons to the Zakharov–Kuznetsov equation, Physica D 43 (1990), 293–303.
Kouranbaeva, S. and Shkoller, S.: A variational approach to second-order multisymplectic field theory, preprint, (1999).
Krupkova, O.: Hamiltonian field theory, J. Geom. Phys. 43 (2002), 93–132.
Marsden, J. E., Patrick, G. W. and Shkoller, S.: Multisymplectic geometry, variational integrators, and nonlinear PDEs, Comm. Math. Phys. 199 (1998), 351–395.
Olver, P. J.: Applications of Lie Groups to Differential Equations, 2nd edn, Springer, New York, 1993.
Reich, S.: Multi-symplectic Runge-Kutta collocation methods for Hamiltonian wave equations, J. Comput. Phys. 157 (2000), 473–499.
Zakharov, V. E. and Kuznetsov, E. A.: Three-dimensional solitons, Soviet. Phys. JEPT 39 (1974), 285–286.
Zhao, P. F. and Qin, M. Z.: Multisymplectic geometry and multisymplectic Preissmann scheme for the KdV equation, J. Phys. A: Math, Gen. 33 (2000), 3613–3626.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Chen, JB. Multisymplectic Geometry, Local Conservation Laws and a Multisymplectic Integrator for the Zakharov–Kuznetsov Equation. Letters in Mathematical Physics 63, 115–124 (2003). https://doi.org/10.1023/A:1023067332646
Issue Date:
DOI: https://doi.org/10.1023/A:1023067332646