zbMATH — the first resource for mathematics

Examples
Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
A multisymplectic integrator for the periodic nonlinear Schrödinger equation. (English) Zbl 1103.65130
Summary: A multisymplectic integrator for the periodic nonlinear Schrödinger equation is presented in this paper. Its accuracy is proved. By introducing a norm, we investigate its nonlinear stability. We also discuss the relationship between this multisymplectic integrator and two variational integrators which are derived by using the discrete multisymplectic field theory and the finite element method.

MSC:
65P10Numerical methods for Hamiltonian systems including symplectic integrators
35Q55NLS-like (nonlinear Schrödinger) equations
37M15Symplectic integrators (dynamical systems)
WorldCat.org
Full Text: DOI
References:
[1] Ablowitz, M. J.; Ladik, J. F.: A nonlinear difference scheme and inverse scattering. Stud. appl. Math 55, 213-229 (1976) · Zbl 0338.35002
[2] Brenner, S. C.; Scott, L. R.: The mathematical theory of finite element methods. (1994) · Zbl 0804.65101
[3] Bridges, T. J.: Multi-symplectic structures and wave propagation. Math. proc. Cam phil. Soc. 121, 147-190 (1997) · Zbl 0892.35123
[4] Bridges, T. J.; Reich, S.: Multi-symplectic integrators: numerical schemes for Hamiltonian pdes that conserve symplecticity. Phys. lett. A 284, 184-193 (2001) · Zbl 0984.37104
[5] Chen, J. B.: Total variation in discrete multisymplectic field theory and multisymplectic energy momentum integrators. Lett. math. Phys. 61, 63-73 (2002) · Zbl 1013.70020
[6] Chen, J. B.; Guo, H. Y.; Wu, K.: Total variation in Hamiltonian formalism and symplectic-energy integrators. J. math. Phys. 44, 1688-1702 (2003) · Zbl 1062.37102
[7] Chen, J. B.; Qin, M. Z.; Tang, Y. F.: Symplectic and multisymplectic methods for the nonlinear Schrödinger equation. Comput. math. Appl. 43, 1095-1106 (2002) · Zbl 1050.65127
[8] Chen, J. B.: A multisymplectic variational integrator for the nonlinear Schrödinger equation. Numer. meth. Part. diff. Eq. 18, 523-536 (2002) · Zbl 1012.65139
[9] J.B. Chen, Variational integrators for discrete multisymplectic field theory based on the finite element method, preprint.
[10] Ciarlet, P. G.: Numerical analysis of the finite element method. (1976) · Zbl 0363.65083
[11] Ciarlet, P. G.: The finite element method for elliptic problems. (1978) · Zbl 0383.65058
[12] Delfour, M.; Fortin, M.; Payre, G.: Finite-difference solutions of a nonlinear Schrödinger equation. J. comput. Phys. 44, 277-288 (1981) · Zbl 0477.65086
[13] Garcia, P. L.: The Poincaré-Cartan invariant in the calculus of variations. Symp. math. 14, 219-246 (1974)
[14] Hasegawa, A.: Optical solitons in fibers. (1989)
[15] Herbst, B. M.; Varadi, F.; Ablowitz, M. J.: Symplectic methods for the nonlinear Schrödinger equation. Math. comput. Simul. 37, 353-369 (1994) · Zbl 0812.65118
[16] J.L. Hong, Y. Liu, H. Munthe-Kass, A. Zanna, On a multisymplectic scheme for Schrödinger equations with variable coefficients, preprint.
[17] Lamb, G. L.: Elements of soliton theory. (1980) · Zbl 0445.35001
[18] Marsden, J. E.; Patrick, G. W.; Shkoller, S.: Multisymplectic geometry, variational integrators, and nonlinear pdes. Comm. math. Phys. 199, 351-395 (1998) · Zbl 0951.70002
[19] Mclachlan, R.: Symplectic integration of Hamiltonian wave equations. Numer. math. 66, 465-492 (1994) · Zbl 0831.65099
[20] Olver, P. J.: Applications of Lie groups to differential equations. (1993) · Zbl 0785.58003
[21] Reich, S.: Multi-symplectic Runge-Kutta collocation methods for Hamiltonian wave equations. J. comput. Phys. 157, 473-499 (2000) · Zbl 0946.65132
[22] Schober, C. M.: Symplectic integrators for Ablowitz-Ladik discrete nonlinear Schrödinger equation. Phys. lett. A 259, 140-151 (1999) · Zbl 0935.37053
[23] Tang, Y. F.; Vázquez, L.; Zhang, F.; Pérez-García, V. M.: Symplectic methods for the nonlinear Schrödinger equation. Comput. math. Appl. 32, 73-83 (1996) · Zbl 0858.65124
[24] Wineberg, S. B.; Mcgrath, J. F.; Gabl, E. F.; Scott, L. R.; Southwell, C. E.: Implicit spectral methods for wave propagation problems. J. comput. Phys. 97, 311-336 (1991) · Zbl 0746.65075