zbMATH — the first resource for mathematics

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.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
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)
Discrete tanh method for nonlinear difference-differential equations. (English) Zbl 1198.65157
Summary: By introducing a simple difference equation to deduce the difference terms and a simple differential equation to deduce the differential terms, we proposed an unified algebraic method for constructing exact solutions to difference-differential equations (DDEs). This method could give many kinds of exact solutions including soliton solutions expressed by hyperbolic functions, periodic solutions expressed by trigonometric functions and rational solutions in a uniform way if solutions of these kinds exist. In this paper, we also give a generalization of the method to determine the degree of DDEs, and compared with the creativity work of D. Baldwin, Ü. Göktas and W. Hereman [Comput. Phys. Comm. 162, 203–217 (2004; Zbl 1196.68324)] through the discrete hybrid equation.
65L99Numerical methods for ODE
[1]Ablowitz, M. J.; Clarkson, P. A.: Nonlinear evolution equations and inverse scattering, (1991) · Zbl 0762.35001
[2]Toda, M.: Nonlinear waves and solitons, (1989)
[3]Toda, M.: Theory of nonlinear lattices, (1981)
[4]Kevrekidis, P. G.; Rasmussen, K. O.; Bishop, A. R.: Int. J. Mod. phys. B, Int. J. Mod. phys. B 15, 2833-2900 (2001)
[5]Tsuchida, T.; Ujino, H.; Wadati, M.: J. phys. A: math. Gen., J. phys. A: math. Gen. 32, 2239-2262 (1999)
[6]Hirota, R.: The direct method in soliton theory, (2004)
[7]Qian, X. M.; Lou, S. Y.; Hu, X. B.: J. phys. A: math. Gen., J. phys. A: math. Gen. 37, 2401-2411 (2004)
[8]Ma, W. X.; Geng, X. G.: CRM proc. Lecture notes, CRM proc. Lecture notes 29, 313-323 (2001)
[9]Ma, W. X.; Fuchssteiner, B.: Int. J. Non-linear mech., Int. J. Non-linear mech. 31, 329-338 (1996)
[10]Fan, E. G.: Phys. lett. A, Phys. lett. A 277, 212-218 (2000)
[11]Baldwin, D.; Göktas, Ü.; Hereman, W.: Comput. phys. Comm., Comput. phys. Comm. 162, 203-217 (2004)
[12]Xie, F. D.; Wang, J. Q.: Chaos, solitons & fractals, Chaos, solitons & fractals 27, 1067-1071 (2006)
[13]Dai, C. Q.; Zhang, J. F.: Chaos, solitons & fractals, Chaos, solitons & fractals 27, 1042-1047 (2006)
[14]Dai, C. Q.; Yang, Q.; Zhang, J. F.: Z. naturforsch. A, Z. naturforsch. A 59, 635-639 (2004)
[15]Ma, Z. Y.; Hu, Y. H.; Lan, J. C.: Chaos, solitons & fractals, Chaos, solitons & fractals 36, 303-308 (2008)
[16]Ma, W. X.; Wu, H. Y.; He, J. S.: Phys. lett. A, Phys. lett. A 364, 29-32 (2007)
[17]Yan, Z. Y.: Non. anal. Theo. meth. Appl., Non. anal. Theo. meth. Appl. 64, No. 8, 1798-1811 (2006)
[18]Hirota, R.; Iwao, M.: Time-discretization of soliton equations, CRM proc. Lecture notes 25, 217-229 (2000) · Zbl 0961.35135
[19]Xie, F. D.; Ji, M.; Zhao, H.: Chaos, solitons & fractals, Chaos, solitons & fractals 33, 1791-1795 (2007)
[20]Lai, X. J.; Zhang, J. F.: Z. naturforsch A, Z. naturforsch A 60, 573-582 (2005)