Exact solutions for some nonlinear evolution equations which describe pseudo-spherical surfaces

Journal of Computational and Applied Mathematics

Volumn 189, Issue 1-2, 2006, Pages 387-411

  Khater, A H ^a,b   Callebaut, D K ^b   Sayed, S M ^a  

^a Beni Suef University   (Egypt)

^b UNIVERSITY OF ANTWERP   (Belgium)

Author keywords

B cklund transformations; Nonlinear evolution equations; Pseudo spherical surfaces

Indexed keywords

COMPUTATION THEORY; MATHEMATICAL TRANSFORMATIONS; SOLITONS; WAVE EQUATIONS;

BÄCKLUND TRANSFORMATIONS; NONLINEAR EVOLUTION EQUATIONS; PSEUDO-SPHERICAL SURFACES;

NONLINEAR EQUATIONS;

EID: 31544468088     PISSN: 03770427     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.cam.2005.10.007     Document Type: Conference Paper

References (50)

1. M.J. Ablowitz, D.J. Kaup, A.C. Newell, and H. Segur The inverse scattering transform-Fourier analysis for nonlinear problems Stud. Appl. Math. 53 1974 249 315
2. R. Beals, M. Rabelo, and K. Tenenblat Bäcklund transformations and inverse scattering solutions for some pseudo-spherical surfaces Stud. Appl. Math. 81 1989 125 151
3. J.A. Cavalcant, and K. Tenenblat Conservation laws for nonlinear evolution equations J. Math. Phys. 29 1988 1044 1049
4. S.S. Chern, and K. Tenenblat Pseudospherical surfaces and evolution equations Stud. Appl. Math. 74 1986 55 83
5. M. Crampin Solitons and SL (2, R ) Phys. Lett. A 66 1978 170 172
6. Q. Ding The NLS-equation and its SL (2, R ) structure J. Phys. A: Math. Gen. 33 2000 325 329
7. Q. Ding, and K. Tenenblat On differential systems describing surfaces of constant curvature J. Differential Equations 184 2002 185 214
8. Q. Ding, and K. Tenenblat Gauge equivalence of differential equations describing surfaces of constant Gaussian curvature Internat. J. Differential Equations Appl. 4 2002 273 284
9. R.K. Dodd, and R.K. Bullough Bäcklund transformations for the AKNS inverse method Phys. Lett. A 62 1977 70 74
10. L.P. Eisenhart, A Treatise on the Differential Geometry of Curves and Surfaces, Ginn & Co., 1909, Dover, New York, 1960.
11. C.S. Gardner, J.M. Greene, M.D. Kruskal, and R.M. Miura Method for solving the Korteweg-de Vries equation Phys. Lett. 19 1967 1095 1097
12. E. Goursat Le Problème de Bäcklund, in: Mémorial des Sciences Mathématiques, Fasc. VI 1925 Gauthier-Villars Paris
13. N. Kamran, and K. Tenenblat On differential equations describing pseudo-spherical surfaces J. Differential Equations 115 1995 75 98
14. A.H. Khater, D.K. Callebaut, A.A. Abdalla, and S.M. Sayed Exact solutions for self-dual Yang-Mills equations Chaos Solitons Fractals 10 1999 1309 1320
15. A.H. Khater, D.K. Callebaut, A.A. Abdalla, A.M. Shehata, and S.M. Sayed Bäcklund transformations and exact solutions for self-dual SU (3 ) Yang-Mills equations Il Nuovo Cimento B 114 1999 1 10
16. A.H. Khater, D.K. Callebaut, and O.H. El-Kalaawy Bäcklund transformations and exact soliton solutions for nonlinear Schrödinger-type equations Il Nuovo Cimento B 113 1998 1121 1136
17. A.H. Khater, D.K. Callebaut, and O.H. El-Kalaawy Bäcklund transformations and exact solutions for a nonlinear elliptic equation modelling isothermal magentostatic atmosphere IMA J. Appl. Math. 65 2000 97 108
18. A.H. Khater, D.K. Callebaut, and R.S. Ibrahim Bäcklund transformations and Painlevé analysis: exact solutions for the unstable nonlinear Schrödinger equation modelling electron-beam plasma Phys. Plasmas 5 1998 395 400
19. A.H. Khater, D.K. Callebaut, and S.M. Sayed Conservation laws for some nonlinear evolution equations which describe pseudospherical surfaces J. Geom. Phys. 51 2004 332 352
20. A.H. Khater, D.K. Callebaut, S.M. Sayed, Bäcklund transformations for some nonlinear evolution equations which describe pseudospherical surfaces, J. Differential Geom., submitted for publication.
21. A.H. Khater, O.H. El-Kalaawy, and D.K. Callebaut Bäcklund transformations for Alfvén solitons in a relativistic electron-positron plasma Phys. Scripta 58 1998 545 548
22. A.H. Khater, M.A. Helal, and O.H. El-Kalaawy Two new classes of exact solutions for the KdV equation via Bäcklund transformations Choas Solitons Fractals 8 1997 1901 1909
23. A.H. Khater, and S.M. Sayed Exact solutions for self-dual SU(2) and SU(3) Yang-Mills fields Internat. J. Theoret. Phys. 41 2002 409 419
24. A.H. Khater, A.M. Shehata, D.K. Callebaut, and S.M. Sayed Self-dual solutions for SU(2) and SU(3) gauge fields one Euclidean space Internat. J. Theoret. Phys. 43 2004 151 159
25. K. Konno, and M. Wadati Simple derivation of Bäcklund transformation from Riccati Form of inverse method Progr. Theoret. Phys. 53 1975 1652 1656
26. M.G. Lamb Bäcklund transformations for certain nonlinear evolution equations J. Math. Phys. 15 1974 2157 2165
27. Y. Lei, Z. Fajiang, and W. Yinghai The homogeneous balance method, Lax pair, Hirota transformation and a general fifth-order KdV equation Choas Solitons Fractals 13 2002 337 340
28. E.J. Parkes, and B.R. Duffy An automated tanh-function method for finding solitary wave solutions to non-linear evolution equations Comput. Phys. Commun. 98 1996 288 300
29. E.G. Reyes Pseudo-spherical surfaces and integrability of evolution equations J. Differential Equations 147 1998 195 230 (Erratum: J. Differential Equations 153 (1999) 223-224.)
30. E.G. Reyes Some geometric aspects of integrability of differential equations in two independent variables Acta Appl. Math. 64 2000 75 109
31. E.G. Reyes Conservation laws and Calapso-Guichard deformations of equations describing pseudo-spherical surfaces J. Math. Phys. 41 2000 2968 2989
32. E.G. Reyes, The soliton content of the Camassa-Holm and Hunter-Saxton equations, Proceedings of the Fourth International Conference on Symmetry in Nonlinear Mathematical Physics, Proceedings of the Institute of Mathematics, NAS Ukraine, 2001.
33. E.G. Reyes Geometric integrability of the Camassa-Holm equation Lett. Math. Phys. 59 2002 117 131
34. E.G. Reyes On generalized Bäcklund transformations for equations describing pseudo-spherical surfaces J. Geom. Phys. 45 2003 368 392
35. E.G. Reyes Transformations of solutions for equations and hierarchies of pseudo-spherical type J. Phys. A: Math. Gen. 36 2003 125 132
36. C. Rogers, and W.K. Schief Bäcklund and Darboux transformations, in: Geometry and Modern Applications in Soliton Theory, Cambridge Texts in Applied Mathematics 2002 Cambridge University Press Cambridge
37. C. Rogers, and W. Shadwick Bäcklund Transformations and their Applications 1982 Academic Press New York
38. R. Sasaki Soliton equation and pseudospherical surfaces Nuclear Phys. B 154 1979 343 357
39. J. Schiff, Symmetries of KdV and Loop Groups, solv-int/9606004 Los Alamos arXiv.org e-Print archive.
40. H.D. Wahlquist, and F.B. Estabrook Bäcklund transformations for solutions of the Korteweg-de Vries equation Phys. Rev. Lett. 31 1973 1386 1390
41. H.D. Wahlquist, and F.B. Estabrook Prolongation structures of nonlinear evolution equations J. Math. Phys. 16 1975 1 7
42. H.D. Wahlquist, and F.B. Estabrook Prolongation structures of nonlinear evolution equations, II J. Math. Phys. 17 1975 1293 1297
43. M.L. Wang Exact solution for a compound KdV-Burgers equation Phys. Lett. A 213 1996 279 287
44. M.L. Wang, and Z.B. Li Application of homogeneous balances method to exact solution of nonlinear equation in mathematical physics Phys. Lett. A 216 1996 67 75
45. W.T. Wu, Polynomial equations-solving and its applications, in: Algorithms and Computation (Beijing 1994), Lecture Notes in Computer Science, vol. 834, Springer, Berlin, 1994, pp. 1-9.
46. T.C. Xia, H.Q. Zhang, and Z.Y. Yan New explicit exact traveling wave solution for a compound KdV-Burgers equation Chinese Phys. 8 2001 694 699
47. C.T. Yan A simple transformation for nonlinear waves Phys. Lett. A 224 1996 77 82
48. T.Z. Yan, and H.Q. Zhang New explicit and exact traveling wave for a system variant Boussinesq equation in mathematical physics Phys. Lett. A 252 1999 291 296
49. V.E. Zakharov, and A.B. Shabat Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media Sov. Phys. JETP 34 1972 62 69
50. X.D. Zhang, T.C. Xia, and H.Q. Zhang New explicit exact traveling wave solution for compound KdV-Burgers equation in mathematical physics Appl. Math. E-Notes 2 2002 45 50