Publication: On the r-th dispersionless Toda hierarchy I: Factorization problem, symmetries and some solutions
Loading...
Official URL
Full text at PDC
Publication Date
2004-08-01
Authors
Advisors (or tutors)
Editors
Journal Title
Journal ISSN
Volume Title
Publisher
IOP Publishing
Citations
Exportar
Abstract
For a family of Poisson algebras, parametrized by r is an element of Z, and an associated Lie algebraic splitting, we consider the factorization of given canonical transformations. In this context, we rederive the recently found rth dispersionless modified KP hierarchies and rth dispersionless Dym hierarchies, giving a new Miura map among them. We also found a new integrable hierarchy which we call the rth dispersionless Toda hierarchy. Moreover, additional symmetries for these hierarchies are studied in detail and new symmetries depending on arbitrary functions are explicitly constructed for the rth dispersionless KP, rth dispersionless Dym and rth dispersionless Toda equations. Some solutions are derived by examining the imposition of a time invariance to the potential rth dispersionless Dym equation, for which a complete integral is presented and, therefore, an appropriate envelope leads to a general solution. Symmetries and Miura maps are applied to get new solutions and solutions of the rth dispersionless modified KP equation.
Description
©IOP Publishing.
The author thanks Luis Martínez Alonso and Elena Medina for several discussions. Partial economical support from Dirección General de Enseñanza Superior e Investigación Científica no. BFM2002-01607 is also acknowledged.
UCM subjects
Unesco subjects
Keywords
Citation
[1] KodamaY and Gibbons J 1990 Integrability of the dispersionless KP hierarchy. Proc. 4thWorkshop on Nonlinear and Turbulent Process in Physics (Singapore: World Scientific)
[2] Kupershmidt B 1985 Commun. Math. Phys. 99 51 Kupershmidt B 1990 J. Phys. A: Math. Gen. 23 871
[3] Tsarev S P 1990 Iz. AN USSR Math. 54 1048
[4] Takasaki K and Takebe T 1995 Rev. Math. Phys. 7 743
[5] Takasaki K and Takebe T 1991 Lett. Math. Phys. 23 205
[6] Takasaki K and Takebe T 1992 Int. J. Mod. Phys. 7 (Suppl. 1B) 889 Takasaki K 2002 Lett. Math. Phys. 59 157
[7] Li L-C 1999 Commun. Math. Phys. 203 573
[8] Krichever I 1994 Commun. Pure. Appl. Math. 47 437 Krichever I 1992 Commun. Math. Phys. 143 415
[9] Dubrovin B 1992 Nucl. Phys. B 342 627 Dubrovin B and Zhang Y 1998 Commun. Math. Phys. 198 311
[10] Gibbons J and Tsarev S P 1989 Phys. Lett. A 135 167 Gibbons J and Tsarev S P 1999 Phys. Lett. A 211 263
[11] Wiegmann P B and Zabrodin A 2000 Commun. Math. Phys. 213 523 Mineev-Weinstein M, Wiegmann P B and Zabrodin A 2000 Phys. Rev. Lett. 84 5106
[12] Konopelchenko B and Martínez Alonso L 2001 Phys. Lett. A 286 161 Konopelchenko B and Martínez Alonso L 2002 J. Math. Phys. 43 3807 Konopelchenko B and Martínez Alonso L 2002 Stud. Appl. Math. 109 313
[13] Mañas M, Martínez Alonso L and Medina E 2002 J. Phys. A: Math. Gen. 335 401
[14] Guil F, Mañas M and Martínez Alonso L 2003 J. Phys. A: Math. Gen. 36 4047
[15] Martínez Alonso L and Mañas M 2003 J. Math. Phys. 44 3294
[16] Guil F, Mañas M and Martínez Alonso L 2003 J. Phys. A: Math. Gen. 36 6457
[17] Ferapontov E V, Korotkin D A and Shramchenko V A 2002 Quantum Grav. 19 L1–L6.M Mañas M and Martínez Alonso L 2002 Preprint nlin.SI/020950 Mañas M and Martínez Alonso L 2003 Theor. Math. Phys. 137 1543 Mañas M and Martínez Alonso L 2004 Phys. Lett. A 320 383
[18] Dunajski M, Mason L J and Tod K P 2001 J. Geom. Phys. 37 63
[19] Dunajski M and Tod K P 2002 Phys. Lett. A 303 253
[20] Ferapontov E V 2002 J. Phys. A: Math. Gen. 35 6883–92 Ferapontov E V and Khusnutdinova K R 2003 On integrability of (2 + 1)- dimensional quasilinear systems Preprint nlin.SI/0305044 Ferapontov E V and Khusnutdinova K R 2003 The characterization of two-component (2 + 1)-dimensional integrable systems of hydrodynamic type Preprint nlin.SI/0310021
[21] Konopelchenko B and Moro A 2004 J. Phys. A: Math. Gen. 37 L105
[22] Błaszak M 2002 Phys. Lett. A 297 191 [23] Błaszak B and Szablikowski B M 2003 J. Phys. A: Math. Gen. 36 12181 Błaszak B and Szablikowski B M 2002 J. Phys. A: Math. Gen. 35 10325, 10345
[24] Golenischeva M and Rieman A G 1988 J. Soviet Math. 169 890
[25] Segal G and Wilson G 1985 Publ. Math. IHES 61 5–65 [26] Chen Y-T and Tu M-H 2003 Lett. Math. Phys. 63 125
[27] Mañas M 2004 S-functions, reductions and hodograph solutions of the r-th dispersionless modified KP and Dym hierarchies Preprint nlin.SI/0405028
[28] Dieudonné J 1975 Élements d’Analyse IV, 19.15 (Paris: Gauthier- Villars)
[29] Mañas M 2004 On the r-th dispersionless Toda hierarchy I: Factorization problem, additional symmetries and some solutions Preprint nlin.SI/0404022