Geodesic
Quasi-Grammian Solutions of the Generalized Coupled Dispersionless Integrable System
Symmetry, integrability and geometry: methods and applications, Tome 8 (2012) Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

The standard binary Darboux transformation is investigated and is used to obtain quasi-Grammian multisoliton solutions of the generalized coupled dispersionless integrable system.
Keywords: integrable systems; binary Darboux transformation; quasideterminants. 
@article{SIGMA_2012_8_a83,
     author = {Bushra Haider and Mahmood-ul Hassan},
     title = {Quasi-Grammian {Solutions} of the {Generalized} {Coupled} {Dispersionless} {Integrable} {System}},
     journal = {Symmetry, integrability and geometry: methods and applications},
     year = {2012},
     volume = {8},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SIGMA_2012_8_a83/}
}
TY  - JOUR
AU  - Bushra Haider
AU  - Mahmood-ul Hassan
TI  - Quasi-Grammian Solutions of the Generalized Coupled Dispersionless Integrable System
JO  - Symmetry, integrability and geometry: methods and applications
PY  - 2012
VL  - 8
UR  - http://geodesic.mathdoc.fr/item/SIGMA_2012_8_a83/
LA  - en
ID  - SIGMA_2012_8_a83
ER  - 
%0 Journal Article
%A Bushra Haider
%A Mahmood-ul Hassan
%T Quasi-Grammian Solutions of the Generalized Coupled Dispersionless Integrable System
%J Symmetry, integrability and geometry: methods and applications
%D 2012
%V 8
%U http://geodesic.mathdoc.fr/item/SIGMA_2012_8_a83/
%G en
%F SIGMA_2012_8_a83
Bushra Haider; Mahmood-ul Hassan. Quasi-Grammian Solutions of the Generalized Coupled Dispersionless Integrable System. Symmetry, integrability and geometry: methods and applications, Tome 8 (2012). http://geodesic.mathdoc.fr/item/SIGMA_2012_8_a83/

[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 ; arXiv: hep-th/9404011 | DOI | MR | Zbl

[2] Carroll R., Kodama Y., “Solution of the dispersionless Hirota equations”, J. Phys. A: Math. Gen., 28 (1995), 6373–6387 ; arXiv: hep-th/9506007 | DOI | MR | Zbl

[3] Cieśliński J., “An algebraic method to construct the Darboux matrix”, J. Math. Phys., 36 (1995), 5670–5706 | DOI | MR | Zbl

[4] Cieśliński 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 ; arXiv: nlin.SI/0506016 | DOI | MR | Zbl

[5] Darboux G., Leçons sur la théorie générale des surfaces et les applications géométriques du calcul infinitésimal, v. IV, D́eformation infiniment petite et représentation sphérique, Gauthier-Villars, Paris, 1896 | Zbl

[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, 28, Springer, Dordrecht, 2007 | MR | Zbl

[8] Dunajski M., “An interpolating dispersionless integrable system”, J. Phys. A: Math. Theor., 41 (2008), 315202, 9 pp. ; arXiv: 0804.1234 | DOI | MR | Zbl

[9] Gel'fand I., Gel'fand S., Retakh V., Wilson R. L., “Quasideterminants”, Adv. Math., 193 (2005), 56–141 ; arXiv: math.QA/0208146 | DOI | MR | Zbl

[10] Gel'fand I., Retakh V., “Determinants of matrices over noncommutative rings”, Funct. Anal. Appl., 25 (1991), 91–102 | DOI | MR

[11] Gel'fand I., Retakh V., “Theory of noncommutative determinants, and characteristic functions of graphs”, Funct. Anal. Appl., 26 (1992), 231–246 | DOI | MR

[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, 210, Amer. Math. Soc., Providence, RI, 2003, 111–123 ; arXiv: math.QA/0206211 | MR | Zbl

[13] Haider B., Hassan M., “Quasideterminant solutions of an integrable chiral model in two dimensions”, J. Phys. A: Math. Theor., 42 (2009), 355211, 18 pp. ; arXiv: 0912.3071 | DOI | MR | Zbl

[14] Haider B., Hassan M., “The ${\mathrm U}(N)$ chiral model and exact multi-solitons”, J. Phys. A: Math. Theor., 41 (2008), 255202, 17 pp. ; arXiv: 0912.1984 | DOI | MR | Zbl

[15] Haider B., Hassan M., Saleem U., “Binary Darboux transformation and quasideterminant solutions of the chiral field”, J. Nonlinear Math. Phys. (to appear) | MR

[16] Hassan M., “Darboux transformation of the generalized coupled dispersionless integrable system”, J. Phys. A: Math. Theor., 42 (2009), 065203, 11 pp. ; arXiv: 0912.1671 | DOI | MR | Zbl

[17] Hirota R., Tsujimoto S., “Note on “New coupled integrable dispersionless equations””, J. Phys. Soc. Japan, 63 (1994), 3533 | DOI

[18] Ji Q., “Darboux transformation for {MZM}-{I}, {II} equations”, Phys. Lett. A, 311 (2003), 384–388 | DOI | MR | Zbl

[19] Kakuhata H., Konno K., “A generalization of coupled integrable, dispersionless system”, J. Phys. Soc. Japan, 65 (1996), 340–341 | DOI | MR | Zbl

[20] Kakuhata H., Konno K., “Canonical formulation of a generalized coupled dispersionless system”, J. Phys. A: Math. Gen., 30 (1997), L401–L407 | DOI | MR | Zbl

[21] Kakuhata H., Konno K., “Lagrangian, Hamiltonian and conserved quantities for coupled integrable, dispersionless equations”, J. Phys. Soc. Japan, 65 (1996), 1–2 | DOI | MR | Zbl

[22] Kodama Y., “A method for solving the dispersionless KP equation and its exact solutions”, Phys. Lett. A, 129 (1988), 223–226 | DOI | MR

[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 | DOI | MR | Zbl

[24] Konno K., Oono H., “New coupled integrable dispersionless equations”, J. Phys. Soc. Japan, 63 (1994), 377–378 | DOI

[25] Konopelchenko B. G., Magri F., “Coisotropic deformations of associative algebras and dispersionless integrable hierarchies”, Comm. Math. Phys., 274 (2007), 627–658 ; arXiv: nlin.SI/0606069 | DOI | MR | Zbl

[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 | DOI | MR | Zbl

[27] Krichever I. M., “The dispersionless Lax equations and topological minimal models”, Comm. Math. Phys., 143 (1992), 415–429 | DOI | MR | Zbl

[28] Krob D., Leclerc B., “Minor identities for quasi-determinants and quantum determinants”, Comm. Math. Phys., 169 (1995), 1–23 ; arXiv: hep-th/9411194 | DOI | MR | Zbl

[29] Leble S. B., “Binary Darboux transformations and systems of $N$ waves in rings”, Theoret. and Math. Phys., 122 (2000), 200–210 | DOI | MR | Zbl

[30] Leble S. B., “Elementary and binary Darboux transformations at rings”, Comput. Math. Appl., 35 (1998), 73–81 | DOI | MR | Zbl

[31] Leble S. B., Ustinov N. V., “Darboux transforms, deep reductions and solitons”, J. Phys. A: Math. Gen., 26 (1993), 5007–5016 | DOI | MR | Zbl

[32] Leble S. B., Ustinov N. V., “Deep reductions for matrix Lax system, invariant forms and elementary Darboux transforms \”, Proceedings of NEEDS-92 Workshop, World Scientific, Singapore, 1993, 34–41 | MR

[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 | DOI | MR | Zbl

[34] Mañas M., “Darboux transformations for the nonlinear Schrödinger equations”, J. Phys. A: Math. Gen., 29 (1996), 7721–7737 | DOI | MR | Zbl

[35] Matveev V. B., Darboux invariance and the solutions of Zakharov–Shabat equations, Preprint LPTHE 79–07, LPTHE, Orsay, 1979, 11 pp.

[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 | DOI | MR | Zbl

[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 | DOI | MR | Zbl

[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 | DOI | MR | Zbl

[39] Matveev V. B., Salle M. A., Darboux transformations and solitons, Springer Series in Nonlinear Dynamics, Springer-Verlag, Berlin, 1991 | MR

[40] Nimmo J. J. C., “Darboux transformations for discrete systems”, Chaos Solitons Fractals, 11 (2000), 115–120 | DOI | MR | Zbl

[41] Nimmo J. J. C., “Darboux transformations from reductions of the KP hierarchy”, Nonlinear Evolution Equations Dynamical Systems, NEEDS'94 (Los Alamos, NM), World Sci. Publ., River Edge, NJ, 1995, 168–177 ; arXiv: solv-int/9410001 | MR | Zbl

[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 | DOI | MR

[43] Oevel W., Schief W., “Darboux theorems and the KP hierarchy”, Applications of Analytic and Geometric Methods to Nonlinear Differential Equations (Exeter, 1992), NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., 413, Kluwer Acad. Publ., Dordrecht, 1993, 193–206 | MR | Zbl

[44] Park Q. H., Shin H. J., “Darboux transformation and Crum's formula for multi-component integrable equations”, Phys. D, 157 (2001), 1–15 | DOI | MR | Zbl

[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 | MR

[46] Sakhnovich A. L., “Dressing procedure for solutions of nonlinear equations and the method of operator identities”, Inverse Problems, 10 (1994), 699–710 | DOI | MR | Zbl

[47] Takasaki K., “Dispersionless Toda hierarchy and two-dimensional string theory”, Comm. Math. Phys., 170 (1995), 101–116 ; arXiv: hep-th/9403190 | DOI | MR | Zbl

[48] Takasaki K., Takebe T., “Integrable hierarchies and dispersionless limit”, Rev. Math. Phys., 7 (1995), 743–808 ; arXiv: hep-th/9405096 | DOI | MR | Zbl

[49] Ustinov N. V., “The reduced self-dual Yang–Mills equation, binary and infinitesimal Darboux transformations”, J. Math. Phys., 39 (1998), 976–985 | DOI | MR | Zbl

[50] Wiegmann P. B., Zabrodin A., “Conformal maps and integrable hierarchies”, Comm. Math. Phys., 213 (2000), 523–538 ; arXiv: hep-th/9909147 | DOI | MR | Zbl