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)
Sine-cosine method for finding the soliton solutions of the generalized fifth-order nonlinear equation. (English) Zbl 1151.35418
Summary: We use a modified form of the sine-cosine method for obtaining exact soliton solutions of the generalized fifth-order nonlinear evolution equation. An analysis of this method is presented. The present method shows that the solutions involve either sec$^{2}$ or sech$^{2}$ under certain conditions. General forms of those conditions are determined for the first time. Exact solutions for special cases of this problem such as the Sawada-Kotera and Lax equations are found.

MSC:
35Q53KdV-like (Korteweg-de Vries) equations
35G20General theory of nonlinear higher-order PDE
WorldCat.org
Full Text: DOI
References:
[1] Ablowitz, M. J.; Clarkson, P. A.: Solitons, nonlinear evolution equations and inverse scattering. (1991) · Zbl 0762.35001
[2] Adomian, G.: Solving frontier problems of physics: the decomposition methods. (1994) · Zbl 0802.65122
[3] Adomian, G.: A review of the decomposition method in applied mathematics. J math anal appl 135, No. 2, 501-544 (1988) · Zbl 0671.34053
[4] Aiyer, R. N.; Fuchssteiner, B.; Oevel, W.: Solitons and discrete eigen-functions of the recursion operator of non-linear evolution equations: I. The caudrey -- dodd -- gibbon -- Sawada-Kotera equations. J phys A: math gen 19, 3755-3770 (1986) · Zbl 0622.35067
[5] Chen, B.; Xie, Y.: An auto-bcklund transformation and exact solutions of stochastic Wick-type Sawada-Kotera equations. Chaos, solitons & fractals 23, No. 1, 243-248 (2005) · Zbl 1064.60148
[6] Hereman, W.; Nuseir, A.: Symbolic methods to construct exact solution of nonlinear differential equations. Math comput simulat 43, No. 1, 13-27 (1997) · Zbl 0866.65063
[7] Fan, E.: Uniformly constructing a series of explicit exact solutions to nonlinear equations in mathematical physics. Chaos, solitons & fractals 16, No. 5, 819-839 (2003) · Zbl 1030.35136
[8] Inc, M.: On numerical soliton solution of the Kaup -- kupershmidt equation and convergences analysis of the decomposition method. Appl math comput 172, No. 1, 72-85 (2006) · Zbl 1088.65089
[9] Musette, M.; Verhoeven, C.: Nonlinear superposition formula for the Kaup -- kupershmidt partial differential equation. Physica D 144, No. 1 -- 2, 211-220 (2000) · Zbl 0961.35145
[10] Parker, A.: On soliton solutions of the Kaup -- kupershmidt equation. II. ’anomalous’ N-soliton solutions. Physica D: Nonlinear phenom 137, No. 1 -- 2, 34-48 (2000) · Zbl 0943.35089
[11] Soliman, A. A.: A numerical simulation and explicit solutions of KdV-Burgers and Lax’s seventh-order KdV equations. Chaos, solitons & fractals 29, No. 2, 294-302 (2006) · Zbl 1099.35521
[12] Syam, M. I.: Adomian decomposition method for approximating the solution of the Korteweg -- devries equation. Appl math comput 162, No. 3, 1465-1473 (2005) · Zbl 1063.65112
[13] Wazwaz, A. M.: A reliable algorithm for obtaining positive solutions for nonlinear boundary value problems. Int J comput math appl 41, 1237-1244 (2001) · Zbl 0983.65090
[14] Wazwaz, A. M.: Approximate solutions to boundary value problems of higher order by the modified decomposition method. Comput math appl 40, 679-691 (2000) · Zbl 0959.65090
[15] Wazwaz, A. M.: Exact solutions for the generalized sine-Gordon and the generalized sinh-Gordon equations. Chaos, solitons & fractals 28, No. 1, 127-135 (2006) · Zbl 1088.35544
[16] Wazwaz, A. M.: The tanh method and a variable separated ODE method for solving double sine-Gordon equation. Phys lett A 350, No. 5 -- 6, 367-370 (2006) · Zbl 1195.65210
[17] Wazwaz, A. M.; Helal, M. A.: Nonlinear variants of the BBM equation with compact and noncompact physical structures. Chaos, solitons & fractals 26, No. 3, 767-776 (2005) · Zbl 1078.35110