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)
Group analysis of KdV equation with time dependent coefficients. (English) Zbl 1197.35231
Summary: We study the generalized KdV equation having time dependent variable coefficients of the damping and dispersion from the Lie group-theoretic point of view. Lie group classification with respect to the time dependent coefficients is performed. The optimal system of one-dimensional subalgebras of the Lie symmetry algebras are obtained. These subalgebras are then used to construct a number of similarity reductions and exact group-invariant solutions, including soliton solutions, for some special forms of the equations.

35Q53KdV-like (Korteweg-de Vries) equations
37K30Relations of infinite-dimensional systems with algebraic structures
35C08Soliton solutions of PDE
35C05Solutions of PDE in closed form
22E70Applications of Lie groups to physics; explicit representations
[1]Biswas, A.: 1-soliton solution of the K(m;n) equation with generalized evolution and time dependent damping and dispersion, Comput. math. Appl. 59, 2536-2540 (2010) · Zbl 1193.35181 · doi:10.1016/j.camwa.2010.01.013
[2]Biswas, A.: Solitary wave solution for the generalized KdV equation with time dependent damping and dispersion, Commun. nonlinear sci. Numer. simul. 14, 3503-3506 (2009) · Zbl 1221.35306 · doi:10.1016/j.cnsns.2008.09.026
[3]Ismail, M. S.; Biswas, A.: 1-soliton solution of the generalized KdV equation with generalized evolution, Appl. math. Comp. 216, 1673-1679 (2010) · Zbl 1190.35200 · doi:10.1016/j.amc.2010.02.045
[4]Wazwaz, A. -M.: New sets of solitary wave solutions to the KdV, mkdv and generalized KdV equations, Commun. nonlinear sci. Numer. simul. 13, 331-339 (2008) · Zbl 1131.35385 · doi:10.1016/j.cnsns.2006.03.013
[5]Xiao-Yan, T.; Fei, H.; Sen-Yue, L.: Variable coefficient KdV equation and the analytic diagnosis of a pole blocking life cycle, Chinese phys. Lett. 23, 887-890 (2006)
[6]Bluman, G. W.; Kumei, S.: Symmetries and differential equations, (1989)
[7], CRC handbook of Lie group analysis of differential equations 1 – 3 (1994)
[8]Olver, P. J.: Applications of Lie groups to differential equations, (1993)
[9]Ovsiannikov, L. V.: Group analysis of differential equations, (1982) · Zbl 0485.58002
[10]Lie, S.: On integration of a class of linear partial differential equations by means of definite integrals, Arch. math. 6, No. 3, 328-368 (1881)
[11]Pakdemirli, M.; Sahin, A. Z.: Group classification of fin equation with variable thermal properties, Int. J. Eng. sci. 42, 1875-1889 (2004) · Zbl 1211.35141 · doi:10.1016/j.ijengsci.2004.04.005
[12]Sophocleous, C.: Further transformation properties of generalized inhomogeneous nonlinear diffusion equations with variable coefficients, Physica A 345, 457-471 (2005)
[13]Liu, H.; Li, J.; Liu, L.: Lie group classifications and exact solutions for two variable coefficient equations, Appl. math. Comp. 215, 2927-2935 (2009) · Zbl 1232.35173 · doi:10.1016/j.amc.2009.09.039
[14]Nadjafikah, M.; Bakhshandeh-Chamazkoti, R.: Symmetry group classification for Burgers equation, Commun. nonlinear sci. Numer. simul. 15, 2303-2310 (2010) · Zbl 1222.35195 · doi:10.1016/j.cnsns.2009.09.031
[15]Senthilkumaran, M.; Pandiaraja, D.; Vaganan, B. Mayil: New exact explicit solutions of the generalized KdV equations, Appl. math. Comput. 202, No. 2, 693-699 (2008) · Zbl 1158.35420 · doi:10.1016/j.amc.2008.03.013
[16], Handbook of mathematical functions (1970)