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)
Canonical reduction of self-dual Yang-Mills equations to Fitzhugh-Nagumo equation and exact solutions. (English) Zbl 1197.35139
Summary: The (constrained) canonical reduction of four-dimensional self-dual Yang-Mills theory to two-dimensional Fitzhugh-Nagumo and the real Newell-Whitehead equations are considered. On the other hand, other methods and transformations are developed to obtain exact solutions for the original two-dimensional Fitzhugh-Nagumo and Newell-Whitehead equations. The corresponding gauge potential $A\mu $ and the gauge field strengths $F\mu \nu $ are also obtained. New explicit and exact traveling wave and solitary solutions (for Fitzhugh-Nagumo and Newell-Whitehead equations) are obtained by using an improved sine-cosine method and the Wu’s elimination method with the aid of Mathematica. Editorial remark: There are doubts about a proper peer-reviewing procedure of this journal. The editor-in-chief has retired, but, according to a statement of the publisher, articles accepted under his guidance are published without additional control.

MSC:
35K55Nonlinear parabolic equations
WorldCat.org
Full Text: DOI
References:
[1] Ward, R. S.: The Painlevé property for the self-dual gauge-field equations, Phys lett A 102, 279-282 (1984)
[2] Belavin, A. A.; Zakharov, V. E.: Yang -- Mills equations as inverse scattering problem, Phys lett B 73, 53-57 (1978)
[3] Mason, L. J.; Newman, E. T.: A connection between the Einstein and Yang -- Mills equations, Commun math phys 121, 659-668 (1989) · Zbl 0668.53048 · doi:10.1007/BF01218161
[4] Woodhouse, N. M.; Mason, L. J.: The Geroch group and non-Hausdorff twistor spaces, Nonlinearity 1, 73-114 (1988) · Zbl 0651.58038 · doi:10.1088/0951-7715/1/1/004
[5] Quantization of the self-dual Yang -- Mills system: exchange algebras and local quantum group in four-dimensional Quantum field theories. 1993;70:1916 -- 19. · Zbl 1051.81536 · doi:10.1103/PhysRevLett.70.1916
[6] Mason, L. J.; Sparling, G. A.: Nonlinear Schrödinger and Korteweg-de Vries are reductions of self-dual Yang -- Mills, Phys lett A 137, 29-33 (1989)
[7] Khater, A. H.; Callebaut, D. K.; Sayed, S. M.: New representation of the self-duality and exact solutions for Yang -- Mills, Int J theor phys 45, 1021-1028 (2006) · Zbl 1100.81033 · doi:10.1007/s10773-006-9095-2
[8] Ward, R. S.: On self-dual gauge fields, Phys lett A 61, 81-82 (1977) · Zbl 0964.81519
[9] Ward, R. S.; Wells, R.: Twistor geometry and field theory, (1990) · Zbl 0729.53068
[10] Khater, A. H.; Callebaut, D. K.; Sayed, S. M.: Conservation laws and exact solutions for some nonlinear partial differential equations, Int J theor phys 45, 589-616 (2006) · Zbl 1106.35083 · doi:10.1007/s10773-006-9048-9
[11] Khater, A. H.; Callebaut, D. K.; Sayed, S. M.: Exact solutions for some nonlinear evolution equations which describe pseudo-spherical surfaces, J comput appl math 189, 387-411 (2006) · Zbl 1093.35005 · doi:10.1016/j.cam.2005.10.007
[12] Ablowitz, M. J.; Clarkson, P. A.: Solitons nonlinear evolution equations and inverse scattering, LMS lecture note series 149 (1992) · Zbl 0762.35001
[13] Gardner, C. S.; Greene, J. M.; Kruskal, M. D.; Miura, R. M.: Method for solving the Korteweg-de Vries equation, Phys rev lett 17, 1095-1097 (1967) · Zbl 1103.35360 · doi:10.1103/PhysRevLett.19.1095
[14] Yang, C. N.; Mills, R. L.: Conservation of isotopic spin and isotopic gauge invariance, Phys rev 96, 191-195 (1954) · Zbl 06538052
[15] Yang, C. N.: Condition of self duality for $SU(2)$ gauge fields on Euclidean four-dimensional space, Phys rev lett 38, 1377-1379 (1977)
[16] Ablowitz, M. J.; Chakravarty, S.; Halburt, R. G.: Integrable systems and reductions of the self-dual Yang -- Mills equations, J math phys 44, 3147-3173 (2003) · Zbl 1062.70050 · doi:10.1063/1.1586967
[17] Yan, C. T.: A simple transformation for nonlinear waves, Phys lett A 224, 77-82 (1996) · Zbl 1037.35504 · doi:10.1016/S0375-9601(96)00770-0
[18] Yan, T. Z.; Zhang, H. Q.: New explicit and exact travelling wave for a system variant Boussinesq equation in mathematical physics, Phys lett A 252, 291-296 (1999) · Zbl 0938.35130 · doi:10.1016/S0375-9601(98)00956-6
[19] Wu, W. T.: Polynomial equations-solving and its applications, Lecture notes in comput sci 834 (1994) · Zbl 0953.01500
[20] Xia, T. C.; Zhang, H. Q.; Yan, Z. Y.: New explicit exact travelling wave solution for a compound KdV-Burgers equation, Chinese phys 8, 694-699 (2001)
[21] Zhang, X. D.; Xia, T. C.; Zhang, H. Q.: New explicit exact travelling wave solution for compound KdV-Burgers equation in mathematical physics, Appl math E-notes 2, 45-50 (2002) · Zbl 0996.35068 · emis:journals/AMEN/2002/2002.html
[22] Wang, M. L.; Li, Z. B.: Application of homogeneous balances method to exact solution of nonlinear equation in mathematical physics, Phys lett A 216, 67-75 (1996) · Zbl 1125.35401 · doi:10.1016/0375-9601(96)00283-6
[23] Parkes, E. J.; Duffy, B. R.: An automated tanh-function method for finding solitary wave solutions to non-linear evolution equations, Comput phys commun 98, 288-300 (1996) · Zbl 0948.76595 · doi:10.1016/0010-4655(96)00104-X
[24] Lei, Y.; Fajiang, Z.; Yinghai, W.: The homogeneous balance method, Lax pair, Hirota transformation and a general fifth-order KdV equation, Choas, solitons & fractals 13, 337-340 (2002) · Zbl 1028.35132 · doi:10.1016/S0960-0779(00)00274-5
[25] Khater, A. H.; Callebaut, D. K.; Sayed, S. M.: Conservation laws for some nonlinear evolution equations which describe pseudospherical surfaces, J geom phys 51, 332-352 (2004) · Zbl 1069.37058 · doi:10.1016/j.geomphys.2003.11.009
[26] Khater, A. H.; Callebaut, D. K.; Abdalla, A. A.; Sayed, S. M.: Exact solutions for self-dual Yang -- Mills equations, Chaos solitons & fractals 10, 1309-1320 (1999) · Zbl 0963.81046
[27] Khater, A. H.; Callebaut, D. K.; Abdalla, A. A.; Shehata, A. M.; Sayed, S. M.: Bäcklund transformations and exact solutions for self-dual $SU(3)$ Yang -- Mills equations, Il nuovo cimento B 144, 1-10 (1999)
[28] Khater, A. H.; Sayed, S. M.: Exact solutions for self-dual $SU(2)$ and $SU(3)$Yang -- Mills fields, Int J theor phys 41, 409-419 (2002) · Zbl 1106.81311 · doi:10.1023/A:1014241120146
[29] Khater, A. H.; Callebaut, D. K.; Shehata, A. M.; Sayed, S. M.: Self-dual solutions for $SU(2)$ and $SU(3)$ gauge fields on Euclidean space, Int J theor phys 43, 151-159 (2004) · Zbl 1058.81053 · doi:10.1023/B:IJTP.0000028857.57274.cd
[30] Fizhugh, R.: Impulse and physiological states in models of nerve membrane, Biophys J 1, 445-466 (1961)
[31] Nagumo, J. S.; Arimoto, S.; Yoshizawa, S.: An active pulse transmission line simulating nerve axon, Proc IRE 50, 2061-2071 (1962)
[32] Aronson, D. G.; Weinberger, H. F.: Multidimensional nonlinear diffusion arising in population genetics, Adv math 30, 33-76 (1978) · Zbl 0407.92014 · doi:10.1016/0001-8708(78)90130-5