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)
Abundant solitons solutions for several forms of the fifth-order KdV equation by using the tanh method. (English) Zbl 1107.65092
Summary: We use the tanh method for solving several forms of the fifth-order nonlinear Korteweg-de Vries (KdV) equation. The forms include those of {\it P. D. Lax} [Commun. Pure Appl. Math. 21, 467--490 (1968; Zbl 0162.41103)], {\it K. Sawada} and {\it T. Kotera} [Prog. Theor. Phys. 51, 1355--1367 (1974; Zbl 1125.35400)], {\it D. Kaup} [Stud. Appl. Math. 62, 189--216 (1980; Zbl 0431.35073)], {\it B. A. Kupershmidt} [A super KdV equation: an integrable system, Phys. Lett. 102A, 213--215 (1984)], {\it M. Ito} [J. Phys. Soc. Jpn. 49, 771--778 (1980)], and other related special cases. Abundant solitons solutions are derived. Two necessary criteria are established to build up reliable strategies that govern the relation between the parameters of the equation. Previously known solutions are recovered and entirely new bell shaped solitons are determined.

65M70Spectral, collocation and related methods (IVP of PDE)
35Q53KdV-like (Korteweg-de Vries) equations
35Q51Soliton-like equations
Full Text: DOI
[1] Goktas, U.; Hereman, W.: Symbolic computation of conserved densities for systems of nonlinear evolution equations. J. symb. Comput. 24, 591-621 (1997) · Zbl 0891.65129
[2] Baldwin, D.; Goktas, U.; Hereman, W.; Hong, L.; Martino, R. S.; Miller, J. C.: Symbolic computation of exact solutions in hyperbolic and elliptic functions for nonlinear pdes. J. symb. Comput. 37, 669-705 (2004) · Zbl 1137.35324
[3] Hereman, W.; Nuseir, A.: Symbolic methods to construct exact solutions of nonlinear partial differential equations. Math. comput. Simul. 43, 13-27 (1997) · Zbl 0866.65063
[4] Ito, M.: An extension of nonlinear evolution equations of the KdV (mKdV) type to higher orders. J. phys. Soc. jpn. 49, 771-778 (1980)
[5] Kaup, D.: On the inverse scattering problem for the cubic eigenvalue problems of the class $\psi $3x+6Q$\psi $x+6R$\psi =\lambda \psi $. Stud. appl. Math. 62, 189-216 (1980) · Zbl 0431.35073
[6] Kupershmidt, B. A.: A super KdV equation: an integrable system. Phys. lett. 102A, 213-215 (1984)
[7] Lax, P. D.: Integrals of nonlinear equations of evolution and solitary waves. Commun. pure appl. Math. 62, 467-490 (1968) · Zbl 0162.41103
[8] Sawada, K.; Kotera, T.: A method for finding N-soliton solutions for the KdV equation and KdV-like equation. Prog. theor. Phys. 51, 1355-1367 (1974) · Zbl 1125.35400
[9] Parker, A.; Dye, J. M.: Boussinesq-type equations and switching solitons. Proc. inst. NAS ukraine 43, No. 1, 344-351 (2002) · Zbl 1034.37040
[10] Hirota, R.: Direct methods in soliton theory. (1980)
[11] Ablowitz, M.; Segur, H.: Solitons and the inverse scattering transform. (1981) · Zbl 0472.35002
[12] Malfliet, W.: Solitary wave solutions of nonlinear wave equations. Am. J. Phys. 60, No. 7, 650-654 (1992) · Zbl 1219.35246
[13] Malfliet, W.: The tanh method: I. Exact solutions of nonlinear evolution and wave equations. Phys. scripta 54, 563-568 (1996) · Zbl 0942.35034
[14] Malfliet, W.: The tanh method: II. Perturbation technique for conservative systems. Phys. scripta 54, 569-575 (1996) · Zbl 0942.35035
[15] Wazwaz, A. M.: The tanh method for travelling wave solutions of nonlinear equations. Appl. math. Comput. 154, No. 3, 713-723 (2004) · Zbl 1054.65106
[16] Wazwaz, A. M.: Partial differential equations: methods and applications. (2002) · Zbl 1079.35001
[17] Wazwaz, A. M.: The tanh method: exact solutions of the sine-Gordon and the sinh-Gordon equations. Appl. math. Comput. 49, 565-574 (2005) · Zbl 1082.65585
[18] Wazwaz, A. M.: Variants of the two-dimensional Boussinesq equations with compactons, solitons and periodic solutions. Comput. math. Appl. 49, 295-301 (2005) · Zbl 1070.35042
[19] Wazwaz, A. M.: The tanh and the sine -- cosine methods for a reliable treatment of the modified equal width equation and its variants. Commun. nonlinear sci. Numer. simul. 112, 148-160 (2006) · Zbl 1078.35108
[20] Wazwaz, A. M.: New compactons, solitons and periodic solutions for nonlinear variants of the KdV and the KP equations. Chaos, solitons compactons 22, No. 1, 249-260 (2004) · Zbl 1062.35121
[21] Wazwaz, A. M.: Special types of the nonlinear dispersive Zakharov -- Kuznetsov equation with compactons, solitons and periodic solutions. Int. J. Comput. math. 81, No. 9, 1107-1119 (2004) · Zbl 1059.35131
[22] Wazwaz, A. M.: A class of nonlinear fourth order variant of a generalized Camassa -- Holm equation with compact and noncompact solutions. Appl. math. Comput. 165, 485-501 (2005) · Zbl 1070.35044