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

SIGMA 12 (2016), 022, 14 pages      arXiv:1512.05817

Hierarchies of Manakov-Santini Type by Means of Rota-Baxter and Other Identities

Błażej M. Szablikowski
Faculty of Physics, Adam Mickiewicz University, Umultowska 85, 61-614 Poznań, Poland

Received January 11, 2016, in final form February 22, 2016; Published online February 27, 2016

The Lax-Sato approach to the hierarchies of Manakov-Santini type is formalized in order to extend it to a more general class of integrable systems. For this purpose some linear operators are introduced, which must satisfy some integrability conditions, one of them is the Rota-Baxter identity. The theory is illustrated by means of the algebra of Laurent series, the related hierarchies are classified and examples, also new, of Manakov-Santini type systems are constructed, including those that are related to the dispersionless modified Kadomtsev-Petviashvili equation and so called dispersionless $r$-th systems.

Key words: Manakov-Santini hierarchy; Rota-Baxter identity; classical $r$-matrix formalism; generalized Lax hierarchies; integrable $(2+1)$-dimensional systems.

pdf (376 kb)   tex (19 kb)


  1. Baxter G., An analytic problem whose solution follows from a simple algebraic identity, Pacific J. Math. 10 (1960), 731-742.
  2. Błaszak M., Classical $R$-matrices on Poisson algebras and related dispersionless systems, Phys. Lett A 297 (2002), 191-195.
  3. Błaszak M., Szablikowski B.M., Classical $R$-matrix theory of dispersionless systems. I. $(1+1)$-dimensional theory, J. Phys. A: Math. Gen. 35 (2002), 10325-10344, nlin.SI/0211008.
  4. Błaszak M., Szablikowski B.M., Classical $R$-matrix theory of dispersionless systems. II. $(2+1)$ dimension theory, J. Phys. A: Math. Gen. 35 (2002), 10345-10364, nlin.SI/0211018.
  5. Błaszak M., Szablikowski B.M., Classical $R$-matrix theory for bi-Hamiltonian field systems, J. Phys. A: Math. Theor. 42 (2009), 404002, 35 pages, arXiv:0902.1511.
  6. Bogdanov L.V., On a class of multidimensional integrable hierarchies and their reductions, Theoret. and Math. Phys. 160 (2009), 887-893, arXiv:0810.2397.
  7. Bogdanov L.V., Non-Hamiltonian generalizations of the dispersionless 2DTL hierarchy, J. Phys. A: Math. Theor. 43 (2010), 434008, 8 pages, arXiv:1003.0287.
  8. Bogdanov L.V., On a class of reductions of the Manakov-Santini hierarchy connected with the interpolating system, J. Phys. A: Math. Theor. 43 (2010), 115206, 11 pages, arXiv:0910.4004.
  9. Bogdanov L.V., Chang J.-H., Chen Y.-T., Generalized dKP: Manakov-Santini hierarchy and its waterbag reduction, arXiv:0810.0556.
  10. Chang J.-H., Chen Y.-T., Hodograph solutions for the Manakov-Santini equation, J. Math. Phys. 51 (2010), 042701, 18 pages, arXiv:0904.4595.
  11. Dunajski M., A class of Einstein-Weyl spaces associated to an integrable system of hydrodynamic type, J. Geom. Phys. 51 (2004), 126-137, nlin.SI/0311024.
  12. Guo L., What is $\ldots$ a Rota-Baxter algebra?, Notices Amer. Math. Soc. 56 (2009), 1436-1437.
  13. Guo L., An introduction to Rota-Baxter algebra, Surveys of Modern Mathematics, Vol. 4, International Press, Somerville, MA, 2012.
  14. Manakov S.V., Santini P.M., Inverse scattering problem for vector fields and the Cauchy problem for the heavenly equation, Phys. Lett. A 359 (2006), 613-619, nlin.SI/0604024.
  15. Manakov S.V., Santini P.M., A hierarchy of integrable partial differential equations in dimension $2+1$, associated with one-parameter families of vector fields, Theoret. and Math. Phys. 152 (2007), 1004-1011.
  16. Manakov S.V., Santini P.M., On the solutions of the dKP equation: the nonlinear Riemann Hilbert problem, longtime behaviour, implicit solutions and wave breaking, J. Phys. A: Math. Theor. 41 (2008), 055204, 23 pages, arXiv:0707.1802.
  17. Manakov S.V., Santini P.M., The dispersionless 2D Toda equation: dressing, Cauchy problem, longtime behaviour, implicit solutions and wave breaking, J. Phys. A: Math. Theor. 42 (2009), 095203, 16 pages, arXiv:0810.4676.
  18. Manakov S.V., Santini P.M., On the solutions of the second heavenly and Pavlov equations, J. Phys. A: Math. Theor. 42 (2009), 404013, 11 pages, arXiv:0812.3323.
  19. Mañas M., On the $r$th dispersionless Toda hierarchy: factorization problem, additional symmetries and some solutions, J. Phys. A: Math. Gen. 37 (2004), 9195-9224, nlin.SI/0404022.
  20. Mañas M., $S$-functions, reductions and hodograph solutions of the $r$th dispersionless modified KP and Dym hierarchies, J. Phys. A: Math. Gen. 37 (2004), 11191-11221, nlin.SI/0405028.
  21. Martínez Alonso L., Shabat A.B., Energy-dependent potentials revisited: a universal hierarchy of hydrodynamic type, Phys. Lett. A 299 (2002), 359-365, nlin.SI/0202008.
  22. Martínez Alonso L., Shabat A.B., Towards a theory of differential constraints of a hydrodynamic hierarchy, J. Nonlinear Math. Phys. 10 (2003), 229-242, nlin.SI/0310036.
  23. Martínez Alonso L., Shabat A.B., Hydrodynamic reductions and solutions of the universal hierarchy, Theoret. and Math. Phys. 140 (2004), 1073-1085, nlin.SI/0312043.
  24. Pavlov M.V., Integrable hydrodynamic chains, J. Math. Phys. 44 (2003), 4134-4156, nlin.SI/0301010.
  25. Pavlov M.V., Chang J.-H., Chen Y.-T., Integrability of the Manakov-Santini hierarchy, arXiv:0910.2400.
  26. Rota G.C., Baxter algebras and combinatorial identities. I, Bull. Amer. Math. Soc. 75 (1969), 325-329.
  27. Rota G.C., Baxter algebras and combinatorial identities. II, Bull. Amer. Math. Soc. 75 (1969), 330-334.
  28. Semenov-Tian-Shansky M.A., Integrable systems and factorization problems, in Factorization and Integrable Systems (Faro, 2000), Oper. Theory Adv. Appl., Vol. 141, Birkhäuser, Basel, 2003, 155-218, nlin.SI/0209057.
  29. Semenov-Tyan-Shanskii M.A., What is a classical $r$-matrix?, Funct. Anal. Appl. 17 (1983), 259-272.
  30. Sergyeyev A., Szablikowski B.M., Central extensions of cotangent universal hierarchy: $(2+1)$-dimensional bi-Hamiltonian systems, Phys. Lett. A 372 (2008), 7016-7023, arXiv:0807.1294.
  31. Szablikowski B.M., Classical $r$-matrix like approach to Frobenius manifolds, WDVV equations and flat metrics, J. Phys. A: Math. Theor. 48 (2015), 315203, 47 pages, arXiv:1304.2075.
  32. Szablikowski B.M., Błaszak M., Meromorphic Lax representations of $(1+1)$-dimensional multi-Hamiltonian dispersionless systems, J. Math. Phys. 47 (2006), 092701, 23 pages, nlin.SI/0510068.
  33. Takasaki K., Takebe T., ${\rm SDiff}(2)$ KP hierarchy, Internat. J. Modern Phys. A 7 (1992), 889-922, hep-th/9112046.
  34. Takasaki K., Takebe T., Integrable hierarchies and dispersionless limit, Rev. Math. Phys. 7 (1995), 743-808, hep-th/9405096.

Previous article  Next article   Contents of Volume 12 (2016)