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

SIGMA 2 (2006), 060, 32 pages      nlin.SI/0606039

q-Deformed KP Hierarchy and q-Deformed Constrained KP Hierarchy

Jingsong He a, b, Yinghua Li a and Yi Cheng a
a) Department of Mathematics, University of Science and Technology of China, Hefei, 230026 Anhui, P.R. China
b) Centre for Scientific Computing, University of Warwick, Coventry CV4 7AL, United Kingdom

Received January 27, 2006, in final form April 28, 2006; Published online June 13, 2006

Using the determinant representation of gauge transformation operator, we have shown that the general form of τ function of the q-KP hierarchy is a q-deformed generalized Wronskian, which includes the q-deformed Wronskian as a special case. On the basis of these, we study the q-deformed constrained KP (q-cKP) hierarchy, i.e. l-constraints of q-KP hierarchy. Similar to the ordinary constrained KP (cKP) hierarchy, a large class of solutions of q-cKP hierarchy can be represented by q-deformed Wronskian determinant of functions satisfying a set of linear q-partial differential equations with constant coefficients. We obtained additional conditions for these functions imposed by the constraints. In particular, the effects of q-deformation (q-effects) in single q-soliton from the simplest τ function of the q-KP hierarchy and in multi-q-soliton from one-component q-cKP hierarchy, and their dependence of x and q, were also presented. Finally, we observe that q-soliton tends to the usual soliton of the KP equation when x 0 and q 1, simultaneously.

Key words: q-deformation; τ function; Gauge transformation operator; q-KP hierarchy; q-cKP hierarchy.

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  1. Klimyk A., Schmüdgen K., q-calculus, in Quantum Groups and Their Represntaions, Berlin, Springer, 1997, Chapter 2, 37-52.
  2. Kac V., Cheung P., Quantum calculus, New York, Springer-Verlag, 2002.
  3. Exton H., q-hypergeometric functions and applications, Chichester, Ellis Horwood Ltd., 1983.
  4. Andrews G.E., q-series: their development and application in analysis, number theory, combinatorics, physics, and computer algebra, Providence, American Mathematical Society, 1986.
  5. Jimbo M., Yang-Baxter equation in integrable systems, Advanced Series in Mathematical Physics, Vol. 10, Singapore, World Scientific, 1990.
  6. Connes A., Noncommutative geometry, San Diego - London, Academic Press, 1994.
  7. Majid S., Free braided differential calculus, braided binomial theorem, and the braided exponential map, J. Math. Phys., 1993, V.34, 4843-4856, hep-th/9302076.
  8. Majid S., Foundations of quantum group theory, Cambridge, Cambridge University Press, 1995, § 10.4.
  9. Zhang D.H., Quantum deformation of KdV hierarchies and their infinitely many conservation laws, J. Phys. A: Math. Gen., 1993, V.26, 2389-2407.
  10. Wu Z.Y., Zhang D.H., Zheng Q.R., Quantum deformation of KdV hierarchies and their exact solutions: q-deformed solitons, J. Phys. A: Math. Gen., 1994, V.27, 5307-5312.
  11. Frenkel E., Reshetikhin N., Quantum affine algebras and deformations of the Virasoro and W-algebras, Comm. Math. Phys., 1996, V.178, 237-264, q-alg/9505025.
  12. Frenkel E., Deformations of the KdV hierarchy and related soliton equations, Int. Math. Res. Not., 1996, V.2, 55-76, q-alg/9511003.
  13. Haine L., Iliev P., The bispectral property of a q-deformation of the Schur polynomials and the q-KdV hierarchy, J. Phys. A: Math. Gen., 1997, V.30, 7217-7227.
  14. Adler M., Horozov E., van Moerbeke P., The solution to the q-KdV equation, Phys. Lett. A, 1998, V.242, 139-151, solv-int/9712015.
  15. Tu M.H., Shaw J.C., Lee C.R., On Darboux-Bäcklund transformations for the q-deformed Korteweg-de Vries hierarchy, Lett. Math. Phys., 1999, V.49, 33-45, solv-int/9811004.
  16. Tu M.H., Shaw J.C., Lee C.R., On the q-deformed modified Korteweg-de Vries hierarchy, Phys. Lett. A, 2000, V.266, 155-159.
  17. Khesin B., Lyubashenko V., Roger C., Extensions and contractions of the Lie algebra of q-pseudodifferential symbols on the circle, J. Funct. Anal., 1997, V.143, 55-97, hep-th/9403189.
  18. Mas J., Seco M., The algebra of q-pseudodifferential symbols and the q-WKP(n) algebra, J. Math. Phys., 1996, V.37, 6510-6529, q-alg/9512025.
  19. Iliev P., Solutions to Frenkel's deformation of the KP hierarchy, J. Phys. A: Math. Gen., 1998, V.31, L241-L244.
  20. Iliev P., Tau function solutions to a q-deformation of the KP hierarchy, Lett. Math. Phys., 1998, V.44, 187-200.
  21. Iliev P., q-KP hierarchy, bispectrality and Calogero-Moser systems, J. Geom. Phys., 2000, V.35, 157-182.
  22. Tu M.H., q-deformed KP hierarchy: its additional symmetries and infinitesimal Bäcklund transformations, Lett. Math. Phys., 1999, V.49, 95-103, solv-int/9811010.
  23. Wang S.K., Wu K., Wu X.N., Wu D.L., The q-deformation of AKNS-D hierarchy, J. Phys. A: Math. Gen., 2001, V.34, 9641-9651.
  24. He J.S., Li Y.H., Cheng Y., q-deformed Gelfand-Dickey hierarchy and the determinant representation of its gauge transformation, Chinese Ann. Math. Ser. A, 2004, V.25, 373-382 (in Chinese).
  25. Konopelchenko B.G., Sidorenko J., Strampp W., (1+1)-dimensional integrable systems as symmetry constraints of (2+1)-dimensional systems, Phys. Lett. A, 1991, V.157, 17-21.
  26. Cheng Y., Li Y.S., The constraint of the Kadomtsev-Petviashvili equation and its special solutions, Phys. Lett. A, 1991, V.157, 22-26.
  27. Oevel W., Strampp W., Constrained KP hierarchy and bi-Hamiltonian structures, Comm. Math. Phys., 1993, V.157, 51-81.
  28. Cheng Y., Constraints of the Kadomtsev-Petviashvili hierarchy, J. Math. Phys., 1992, V.33, 3774-3782.
  29. Cheng Y., Modifying the KP, the nth constrained KP hierarchies and their Hamiltonian structures, Comm. Math. Phys., 1995, V.171, 661-682.
  30. Aratyn H., Ferreira L.A., Gomes J.F., Zimerman A.H., Constrained KP models as integrable matrix hierarchies, J. Math. Phys., 1997, V.38, 1559-1576, hep-th/9509096.
  31. Aratyn H., On Grassmannian description of the constrained KP hierarchy, J. Geom. Phys., 1999, V.30, 295-312, solv-int/9805006.
  32. Chau L.L., Shaw J.C., Yen H.C., Solving the KP hierarchy by gauge transformations, Comm. Math. Phys., 1992, V.149, 263-278.
  33. Oevel W., Rogers C., Gauge transformations and reciprocal links in 2+1 dimensions, Rev. Math. Phys., 1993, V.5, 299-330.
  34. He J.S., Li Y.S., Cheng Y., The determinant representation of the gauge transformation operators, Chinese Ann. Math. Ser. B, 2002, V.23, 475-486.
  35. He J.S., Li Y.S., Cheng Y., Two choices of the gauge transformation for the AKNS hierarchy through the constrained KP hierarchy, J. Math. Phys., 2003, V.44, 3928-3960.
  36. Date E., Kashiwara M., Jimbo M., Miwa T., Transformation group for soliton equations, in Bosonization, Editor M. Stone, Singapore, World Scientific, 1994, 427-507.
  37. Dickey L.A., Soliton equations and Hamiltonian systems, Singapore, World Scientific, 1991.
  38. Oevel W., Strampp W., Wronskian solutions of the constrained Kadomtsev-Petviashvili hierarchy, J. Math. Phys., 1996, V.37, 6213-6219.
  39. Ohta Y., Satsuma J., Takahashi D., Tokihiro T., An elementary introduction to Sato theory, Progr. Theoret. Phys. Suppl., 1988, N 94, 210-241.
  40. Ablowitz M.J., Clarkson P.A., Solitons, nonlinear evolution equations and inverse scattering, Cambridge, Cambridge University Press, 1991.

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