Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 8 (2012), 084, 15 pages      arXiv:1211.1762

Quasi-Grammian Solutions of the Generalized Coupled Dispersionless Integrable System

Bushra Haider and Mahmood-ul Hassan
Department of Physics, University of the Punjab, Quaid-e-Azam Campus, Lahore-54590, Pakistan

Received June 22, 2012, in final form October 10, 2012; Published online November 08, 2012

The standard binary Darboux transformation is investigated and is used to obtain quasi-Grammian multisoliton solutions of the generalized coupled dispersionless integrable system.

Key words: integrable systems; binary Darboux transformation; quasideterminants.

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  1. Aoyama S., Kodama Y., Topological conformal field theory with a rational W potential and the dispersionless KP hierarchy, Modern Phys. Lett. A 9 (1994), 2481-2492, hep-th/9404011.
  2. Carroll R., Kodama Y., Solution of the dispersionless Hirota equations, J. Phys. A: Math. Gen. 28 (1995), 6373-6387, hep-th/9506007.
  3. Cieslinski J., An algebraic method to construct the Darboux matrix, J. Math. Phys. 36 (1995), 5670-5706.
  4. Cieslinski J., Biernacki W., A new approach to the Darboux-Bäcklund transformation versus the standard dressing method, J. Phys. A: Math. Gen. 38 (2005), 9491-9501, nlin.SI/0506016.
  5. Darboux G., Leçons sur la théorie générale des surfaces et les applications géométriques du calcul infinitésimal. Vol. IV. Deformation infiniment petite et représentation sphérique, Gauthier-Villars, Paris, 1896.
  6. Darboux G., Sur une proposition relative aux équations linéaires, C. R. Acad. Sci. Paris 94 (1882), 1456-1459.
  7. Doktorov E.V., Leble S.B., A dressing method in mathematical physics, Mathematical Physics Studies, Vol. 28, Springer, Dordrecht, 2007.
  8. Dunajski M., An interpolating dispersionless integrable system, J. Phys. A: Math. Theor. 41 (2008), 315202, 9 pages, arXiv:0804.1234.
  9. Gel'fand I., Gel'fand S., Retakh V., Wilson R.L., Quasideterminants, Adv. Math. 193 (2005), 56-141, math.QA/0208146.
  10. Gel'fand I., Retakh V., Determinants of matrices over noncommutative rings, Funct. Anal. Appl. 25 (1991), 91-102.
  11. Gel'fand I., Retakh V., Theory of noncommutative determinants, and characteristic functions of graphs, Funct. Anal. Appl. 26 (1992), 231-246.
  12. Gel'fand I., Retakh V., Wilson R.L., Quaternionic quasideterminants and determinants, in Lie Groups and Symmetric Spaces, Amer. Math. Soc. Transl. Ser. 2, Vol. 210, Amer. Math. Soc., Providence, RI, 2003, 111-123, math.QA/0206211.
  13. Haider B., Hassan M., Quasideterminant solutions of an integrable chiral model in two dimensions, J. Phys. A: Math. Theor. 42 (2009), 355211, 18 pages, arXiv:0912.3071.
  14. Haider B., Hassan M., The U(N) chiral model and exact multi-solitons, J. Phys. A: Math. Theor. 41 (2008), 255202, 17 pages, arXiv:0912.1984.
  15. Haider B., Hassan M., Saleem U., Binary Darboux transformation and quasideterminant solutions of the chiral field, J. Nonlinear Math. Phys., to appear.
  16. Hassan M., Darboux transformation of the generalized coupled dispersionless integrable system, J. Phys. A: Math. Theor. 42 (2009), 065203, 11 pages, arXiv:0912.1671.
  17. Hirota R., Tsujimoto S., Note on "New coupled integrable dispersionless equations", J. Phys. Soc. Japan 63 (1994), 3533.
  18. Ji Q., Darboux transformation for MZM-I, II equations, Phys. Lett. A 311 (2003), 384-388.
  19. Kakuhata H., Konno K., A generalization of coupled integrable, dispersionless system, J. Phys. Soc. Japan 65 (1996), 340-341.
  20. Kakuhata H., Konno K., Canonical formulation of a generalized coupled dispersionless system, J. Phys. A: Math. Gen. 30 (1997), L401-L407.
  21. Kakuhata H., Konno K., Lagrangian, Hamiltonian and conserved quantities for coupled integrable, dispersionless equations, J. Phys. Soc. Japan 65 (1996), 1-2.
  22. Kodama Y., A method for solving the dispersionless KP equation and its exact solutions, Phys. Lett. A 129 (1988), 223-226.
  23. Kodama Y., Pierce V.U., Combinatorics of dispersionless integrable systems and universality in random matrix theory, Comm. Math. Phys. 292 (2009), 529-568, arXiv:0811.0351.
  24. Konno K., Oono H., New coupled integrable dispersionless equations, J. Phys. Soc. Japan 63 (1994), 377-378.
  25. Konopelchenko B.G., Magri F., Coisotropic deformations of associative algebras and dispersionless integrable hierarchies, Comm. Math. Phys. 274 (2007), 627-658, nlin.SI/0606069.
  26. Kotlyarov V.P., On equations gauge equivalent to the sine-Gordon and Pohlmeyer-Lund-Regge equations, J. Phys. Soc. Japan 63 (1994), 3535-3537.
  27. Krichever I.M., The dispersionless Lax equations and topological minimal models, Comm. Math. Phys. 143 (1992), 415-429.
  28. Krob D., Leclerc B., Minor identities for quasi-determinants and quantum determinants, Comm. Math. Phys. 169 (1995), 1-23, hep-th/9411194.
  29. Leble S.B., Binary Darboux transformations and systems of N waves in rings, Theoret. and Math. Phys. 122 (2000), 200-210.
  30. Leble S.B., Elementary and binary Darboux transformations at rings, Comput. Math. Appl. 35 (1998), 73-81.
  31. Leble S.B., Ustinov N.V., Darboux transforms, deep reductions and solitons, J. Phys. A: Math. Gen. 26 (1993), 5007-5016.
  32. Leble S.B., Ustinov N.V., Deep reductions for matrix Lax system, invariant forms and elementary Darboux transforms, in Proceedings of NEEDS-92 Workshop, World Scientific, Singapore, 1993, 34-41.
  33. Li C.X., Nimmo J.J.C., Quasideterminant solutions of a non-abelian Toda lattice and kink solutions of a matrix sine-Gordon equation, Proc. R. Soc. Lond. Ser. A 464 (2008), 951-966, arXiv:0711.2594.
  34. Mañas M., Darboux transformations for the nonlinear Schrödinger equations, J. Phys. A: Math. Gen. 29 (1996), 7721-7737.
  35. Matveev V.B., Darboux invariance and the solutions of Zakharov-Shabat equations, Preprint LPTHE 79-07, LPTHE, Orsay, 1979, 11 pages.
  36. Matveev V.B., Darboux transformation and explicit solutions of the Kadomtcev-Petviaschvily equation, depending on functional parameters, Lett. Math. Phys. 3 (1979), 213-216.
  37. Matveev V.B., Darboux transformation and the explicit solutions of differential-difference and difference-difference evolution equations. I, Lett. Math. Phys. 3 (1979), 217-222.
  38. Matveev V.B., Salle M.A., Differential-difference evolution equations. II. Darboux transformation for the Toda lattice, Lett. Math. Phys. 3 (1979), 425-429.
  39. Matveev V.B., Salle M.A., Darboux transformations and solitons, Springer Series in Nonlinear Dynamics, Springer-Verlag, Berlin, 1991.
  40. Nimmo J.J.C., Darboux transformations for discrete systems, Chaos Solitons Fractals 11 (2000), 115-120.
  41. Nimmo J.J.C., Darboux transformations from reductions of the KP hierarchy, in Nonlinear Evolution Equations & Dynamical Systems: NEEDS '94 (Los Alamos, NM), World Sci. Publ., River Edge, NJ, 1995, 168-177, solv-int/9410001.
  42. Nimmo J.J.C., Gilson C.R., Ohta Y., Applications of Darboux transformations to the self-dual Yang-Mills equations, Theoret. and Math. Phys. 122 (2000), 239-246.
  43. Oevel W., Schief W., Darboux theorems and the KP hierarchy, in Applications of Analytic and Geometric Methods to Nonlinear Differential Equations (Exeter, 1992), NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., Vol. 413, Kluwer Acad. Publ., Dordrecht, 1993, 193-206.
  44. Park Q.H., Shin H.J., Darboux transformation and Crum's formula for multi-component integrable equations, Phys. D 157 (2001), 1-15.
  45. Rogers C., Schief W.K., Bäcklund and Darboux transformations: geometry and modern applications in soliton theory, Cambridge Texts in Applied Mathematics, Cambridge University Press, Cambridge, 2002.
  46. Sakhnovich A.L., Dressing procedure for solutions of nonlinear equations and the method of operator identities, Inverse Problems 10 (1994), 699-710.
  47. Takasaki K., Dispersionless Toda hierarchy and two-dimensional string theory, Comm. Math. Phys. 170 (1995), 101-116, hep-th/9403190.
  48. Takasaki K., Takebe T., Integrable hierarchies and dispersionless limit, Rev. Math. Phys. 7 (1995), 743-808, hep-th/9405096.
  49. Ustinov N.V., The reduced self-dual Yang-Mills equation, binary and infinitesimal Darboux transformations, J. Math. Phys. 39 (1998), 976-985.
  50. Wiegmann P.B., Zabrodin A., Conformal maps and integrable hierarchies, Comm. Math. Phys. 213 (2000), 523-538, hep-th/9909147.

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