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)
Some new travelling wave solutions with singular or nonsingular character for the higher order wave equation of KdV type (III). (English) Zbl 1167.34006
Summary: The integral bifurcation method was used to study the higher order nonlinear wave equations of KdV type (III), which was first proposed by Fokas. Some new travelling wave solutions with singular or nonsingular character are obtained. In particular, we obtain a peculiar exact solution of parametric type in this paper. This solution has three kinds of wave-form including solitary wave, cusp wave and loop solion under different wave velocity conditions. This phenomenon has proved that the loop soliton solution is one continuous solution, not three breaking solutions.

MSC:
34B40Boundary value problems for ODE on infinite intervals
34A05Methods of solution of ODE
35Q53KdV-like (Korteweg-de Vries) equations
35Q51Soliton-like equations
WorldCat.org
Full Text: DOI
References:
[1] Tzirtzilakis, E.; Marinakis, V.; Apokis, C.; Bountis, T.: Soliton-like solutions of higher order wave equations of the Korteweg-de-Vries type. Journal of mathematical physics 43, No. 12, 6151-6161 (2002) · Zbl 1060.35127
[2] Tzirtzilakis, E.; Xenos, M.; Marinakis, V.; Bountis, T.: Interactions and stability of solitary waves in shallow water. Chaos, solitons and fractals 14, 87-95 (2002) · Zbl 1068.76011
[3] Fokas, A.: On a class of physically important integral equations. Physica D 87, 145-150 (1995) · Zbl 1194.35363
[4] Long, Y.; Rui, W.; He, B.: Travelling wave solutions for a higher order wave equations of KdV type (I). Chaos, solitons and fractals 23, 469-475 (2005) · Zbl 1069.35075
[5] Khuri, S. A.: Soliton and periodic solutions for higher order wave equations of KdV type (I). Chaos, solitons and fractals 26, 25-32 (2005) · Zbl 1070.35062
[6] Long, Y.; He, B.; Rui, W.; Chen, C.: Compacton-like and kink-like waves for a higher-order wave equation of Korteweg-de Vries type. International journal of computer mathematics 83, No. 12, 959-971 (2006) · Zbl 1134.35096
[7] Rui, W.; Long, Y.; He, B.: Periodic wave solutions and solitary cusp wave solutions for a higher order wave equation of KdV type. Rostocker mathematisches kolloquium 61, 56-70 (2005)
[8] Li, J.; Rui, W.; Long, Y.; He, B.: Travelling wave solutions for a higher order wave equation of KdV type (III). Mathemmatical biosciences and engineering 3, No. 1, 125-135 (2006)
[9] Long, Y.; Li, J.; Rui, W.; He, B.: Travelling wave solutions for a higher order wave equation of KdV type. Applied mathematics and mechanics 28, No. 11, 1455-1465 (2007) · Zbl 1231.35035
[10] Rui, W.; He, B.; Long, Y.; Chen, C.: The integral bifurcation method and its application for solving a family of third-order dispersive pdes. Nonlinear analysis (2007) · Zbl 1144.35461
[11] Rui, W.; Xie, S.; He, B.; Long, Y.: Integral bifurcation method and its application for solving the modified equal width wave equation and its variants. Rostocker mathematisches kolloquium 62, 87-106 (2007) · Zbl 1148.35079
[12] Li, J.; Liu, Z.: Smooth and non-smooth travelling waves in a nonlinearly dispersive equation. Applied mathematical modelling 25, 41-56 (2000) · Zbl 0985.37072
[13] He, B.; Meng, Q.; Rui, W.; Long, Y.: Bifurcations of travelling wave solutions for the $mK(n,n)$ equation. Communications in nonlinear science and numerical simulation (2007)
[14] He, B.; Li, J.; Long, Y.; Rui, W.: Bifurcations of travelling wave solutions for a variant of Camassa--Holm equation. Nonlinear analysis: real world applications 9, 222-232 (2008) · Zbl 1185.35217
[15] Meng, Q.; He, B.; Long, Y.; Rui, W.: Bifurcations of travelling wave solutions for a general sine-Gordon equation. Chaos, solitons and fractals 29, 483-489 (2006) · Zbl 1099.35116
[16] Liu, Z.; Long, Y.: Compacton-like wave and kink-like wave of GCH equation. Nonlinear analysis: real world applications 8, 136-155 (2007) · Zbl 1106.35065
[17] Yomba, E.: The extended F-expansion method and its application for solving the nonlinear wave, CKGZ,GDS,DS GZ equations. Physics letters A 340, 149-160 (2005) · Zbl 1145.35455
[18] Rui, W.; He, B.; Long, Y.: The binary F-expansion method and its application for solving the (n+1)-dimensional sine-Gordon equation. Communications in nonlinear science and numerical simulation (2008) · Zbl 1221.35360
[19] Abdou, M.: The extended F-expansion method and its application for a class of nonlinear evolution equations. Chaos, solitons and fractals 31, 95-104 (2007) · Zbl 1138.35385
[20] Rui, W.; Long, Y.; He, B.; Li, Z.: Integral bifurcation method combined with computer for solving a higher order wave equation of KdV type. International journal of computer mathematics (2008) · Zbl 1182.65161
[21] Bressan, A.; Constantin, A.: Global conservative solutions of the Camassa--Holm equation. Archive for rational mechanics and analysis 183, 215-239 (2007) · Zbl 1105.76013
[22] Constantin, A.; Strauss, W. A.: Stability of peakons. Communications on pure and applied mathematics 53, 303-310 (2000) · Zbl 1049.35149
[23] Victor, K. K.; Thomas, B. B.; Timoleon, C. K.: Reply to comment on ’on two-loop soliton solution of the Schäfer--Wayne short-pulse equation using Hirota’s method and Hodnett-Moloney approach’. Journal of the physical society of Japan 76, No. 11, 116002-1-2 (2007)
[24] Zhang, Y.; Li, J.: Comment on ”on two-loop soliton solution of the Schäfer--Wayne short-pulse equation using Hirota’s method and Hodnett-molony approach”. Journal of the physical society of Japan 76, No. 11, 116001-1-2 (2007)