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)
Travelling wave solutions of nonlinear partial equations by using the first integral method. (English) Zbl 1191.35090
Summary: The first integral method is employed for constructing the new exact travelling wave solutions of nonlinear partial differential equations. The power of this manageable method is confirmed by applying it for two selected nonlinear partial equations. This approach can also be applied to other systems of nonlinear differential equations.
MSC:
35C07Traveling wave solutions of PDE
35C05Solutions of PDE in closed form
35A25Other special methods (PDE)
References:
[1]Ablowitz, M.; Clarkson, P. A.: Soliton nonlinear evolution equations and inverse scattering, (1991) · Zbl 0762.35001
[2]Wadati, M.: Wave propagation in nonlinear lattice, J. phys. Soc. jpn. 38, 673 (1975)
[3]Miura, M. R.: Bäcklund transformation, (1978)
[4]Matveev, V. A.; Salle, M. A.: Darboux transformation and solitons, (1991) · Zbl 0744.35045
[5]Gu, C. H.: Darboux transformation in solitons theory and geometry applications, (1999)
[6]Hirota, Ryogo: Exact solution of the Korteweg – de Vries equation for multiple collisions of solitons, Phys. rev. Lett. 27, 192-1194 (1971) · Zbl 1168.35423 · doi:10.1103/PhysRevLett.27.1192
[7]Hirota, Ryogo: The direct method in soliton theory, (2004)
[8]Wang, M. L.: Solitary wave solutions for variant Boussinesq equations, Phys. lett. A. 199, 169-172 (1995) · Zbl 1020.35528 · doi:10.1016/0375-9601(95)00092-H
[9]Fan, E. G.; Zhang, H. Q.: A note on the homogeneous balance method, Phys. lett. A 246, 403-406 (1998) · Zbl 1125.35308 · doi:10.1016/S0375-9601(98)00547-7
[10]Cheng, Y.; Wang, Q.: A new general algebraic method with symbolic computation to construct new travelling wave solution for the (1+1)-dimensional dispersive long wave equation, Appl. math. Comput. 168, 1189-1204 (2005) · Zbl 1082.65578 · doi:10.1016/j.amc.2004.10.012
[11]Fan, E. G.: Extended tanh-function method and its applications to nonlinear equations, Phys. lett. A. 277, 212-218 (2000) · Zbl 1167.35331 · doi:10.1016/S0375-9601(00)00725-8
[12]Yan, Z. Y.: New explicit travelling wave solutions for two new integrable coupled nonlinear evolution equations, Phys. lett. A. 292, 100-106 (2001) · Zbl 1092.35524 · doi:10.1016/S0375-9601(01)00772-1
[13]Fan, E. G.: Travelling wave solutions in terms of special functions for nonlinear coupled evolution systems, Phys. lett. A. 300, 43-249 (2002) · Zbl 0997.34007 · doi:10.1016/S0375-9601(02)00776-4
[14]Fan, E. G.: A new algebraic method for finding the line soliton solutions and doubly periodic wave solution to a two-dimensional perturbed KdV equation, Chaos solitons fract. 15, 67-574 (2003) · Zbl 1037.76049 · doi:10.1016/S0960-0779(02)00146-7
[15]Abdou, M. A.: An improved generalized F-expansion method and its applications, J. comput. Appl. math. 214, 202-208 (2008) · Zbl 1135.65378 · doi:10.1016/j.cam.2007.02.030
[16]Yomba, Emmanuel: The modified extended Fan sub-equation method and its application to the (2+1)-dimensional Broer – Kaup – kupershmidt equation, Chaos solitons fract. 27, 187-196 (2006) · Zbl 1088.35532 · doi:10.1016/j.chaos.2005.03.021
[17]Feng, Z. S.: The first integral method to study the Burgers – KdV equation, J. phys. A: math. Gen. 35, 343-349 (2002) · Zbl 1040.35096 · doi:10.1088/0305-4470/35/2/312
[18]Feng, Z. S.; Wang, X. H.: The first integral method to the two-dimensional Burgers – KdV equation, Phys. lett. A 308, 173-178 (2003) · Zbl 1008.35062 · doi:10.1016/S0375-9601(03)00016-1
[19]Feng, Z. S.: Traveling wave behavior for a generalized Fisher equation, Chaos solitons fract. 38, 481-488 (2008) · Zbl 1146.35380 · doi:10.1016/j.chaos.2006.11.031
[20]Raslan, K. R.: The first integral method for solving some important nonlinear partial differential equations, Nonlinear dynam. 53, 281 (2008) · Zbl 1176.35149 · doi:10.1007/s11071-007-9262-x
[21]Bourbaki, N.: Commutative algebra, (1972)
[22]Lou, S. Y.; Hu, X. B.: Infinitely many Lax pairs and symmetry constraints of the KP equation, J. math. Phys. 38, 6401-6427 (1997) · Zbl 0898.58029 · doi:10.1063/1.532219
[23]Ruan, H. Y.; Chen, Y. X.: Study of a (2+1)-dimensional Broer – Kaup equation, Acta phys. Sinica (Overseas ed.) 7, 241-248 (1998)
[24]Lou, S. Y.: (2+1)-dimensional compacton solutions with and without completely elastic interaction properties, J. phys. A: math. Gen. 35, 10619-10628 (2002) · Zbl 1040.35101 · doi:10.1088/0305-4470/35/49/310
[25]Zhi, H. Y.; Wang, Q.; Zhang, H. Q.: A series of new exact solutions to the (2+1)-dimensional Broer – kau – kupershmidt equation, Acta phys. Sinica 54, No. 3, 1002-1007 (2005) · Zbl 1202.35291
[26]Wan, Y.; Song, L. N.; Y., L.; Zhang, H. Q.: Generalized method and new exact wave solutions for (2+1)-dimensional Broer – Kaup – kupershmidt system, Appl. math. Comput 187, 644-657 (2007) · Zbl 1114.65353 · doi:10.1016/j.amc.2006.08.082
[27]K. Weierstrass, Mathematische Werke, vol. V, Johnson, New York, 1915.
[28]Whittaker, E. T.; Watson, G. N.: A course of modern analysis, (1927) · Zbl 53.0180.04
[29]Chandrasekharan, K.: Elliptic functions, (1985)
[30]Schürmann, H. W.: Travelling wave solutions for cubic – quintic nonlinear Schrödinger equation, Phys. rev. E 54, 4312-4320 (1996)
[31]Abramovitz, M.; Stegun, I. A.: Handbook of mathematical functions, (1972)
[32]Ping, Y. J.; Yue, L. S.: Multilinear variable separation approach in (3+1)-dimensions: the Burgers equation, Chin. phys. Lett. 20, 1448-1451 (2003)
[33]Peng, S. S.; Liang, P. Z.; Jun, Z.: New exact solution to (3+1)-dimensional Burgers equation, Commun. theor. Phys. 42, 49-50 (2004) · Zbl 1167.35486
[34]Wang, Q.; Chen, Y.; Zhang, H.: A new Riccati equation rational expansion method and its application to (2+1)-dimensional Burgers equation, Chaos solitons fract. 25, 1019-1028 (2005) · Zbl 1070.35073 · doi:10.1016/j.chaos.2005.01.039
[35]Tang, X. Y.; Lou, S.: Variable separation solutions for the (2+1)-dimensional Burgers equation, Chin. phys. Lett. 20, 335-337 (2003)
[36]Wazwaz, A. M.: Multiple soliton solutions and multiple singular soliton solutions for the (3+1)-dimensional Burgers equations, Appl. math. Comput. 200, 942-948 (2008) · Zbl 1154.65367 · doi:10.1016/j.amc.2008.08.004