KP Solitons, Higher Bruhat and Tamari Orders

  • Aristophanes DimakisEmail author
  • Folkert Müller-Hoissen
Part of the Progress in Mathematics book series (PM, volume 299)


In a tropical approximation, any tree-shaped line soliton solution, a member of the simplest class of soliton solutions of the Kadomtsev-Petviashvili (KP-II) equation, determines a chain of planar rooted binary trees, connected by right rotation. More precisely, it determines a maximal chain of a Tamari lattice. We show that an analysis of these solutions naturally involves higher Bruhat and higher Tamari orders.


Linear Extension Maximal Chain Hasse Diagram Bruhat Order Tetrahedron Equation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. 1.
    V. Bazhanov and Y. Stroganov, “Conditions of commutativity of transfer matrices on a multidimensional lattice”, Theor. Math. Phys. 52 (1982) 685-691.MathSciNetCrossRefGoogle Scholar
  2. 2.
    A. Dimakis and F. Mtiller-Hoissen, “KP line solitons and Tamari lattices”, J. Phys. A: Math. Theor. 44 (2011)025203.CrossRefGoogle Scholar
  3. 3.
    P. Edelman and V. Reiner, “The higher Tamari posets”, Mathematika 43 (1996) 127-154.MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    S. Felsner and H. Weil, “A theorem on higher Bruhat orders”, Discr. Comput. Geom. 23 (2000) 121-127.MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    S. Felsner and G. Ziegler, “Zonotopes associated with higher Bruhat orders”, Discr. Math. 241 (2001) 301-312.MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    I. Frenkel and G. Moore, “Simplex equations and their solutions”, Commun. Math. Phys. 138 (1991) 259-271.MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    J. Hietarinta and F. Nijhoff, “The eight tetrahedron equations”, J. Math. Phys. 38 (1997) 3603-3615.Google Scholar
  8. 8.
    M. Kapranov and V. Voevodsky, “Combinatorial-geometric aspects of polycategory theory: pasting schemes and higher bruhat orders (list of results)”, Cahiers de topologie et géométrie différentielle catégoriques 32 (1991) 11-27.MathSciNetzbMATHGoogle Scholar
  9. 9.
    Y. Kodama, “KP solitons in shallow water”, J. Phys. A: Math. Theor. 43 (2010) 434004.MathSciNetCrossRefGoogle Scholar
  10. 10.
    I. Korepanov, “Tetrahedral Zamolodchikov algebras corresponding to Baxter’s L-operators”, Comm. Math. Phys. 154 (1993) 85-97.MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    _____, “The tetrahedron equation and algebraic geometry”, J. Math. Sci. 83 (1997) 85-92.MathSciNetCrossRefGoogle Scholar
  12. 12.
    R. Lawrence, “Algebras and triangle relations”, J. Pure Appl. Alg. 100 (1995) 43-72.MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    _____, “Yang-Baxter type equations and posets of maximal chains”, J. Comb. Theory, Ser. A 79 (1997) 68-104.Google Scholar
  14. 14.
    J.-L. Loday, “Dichotomy of the addition of natural numbers”, in this volume. Google Scholar
  15. 15.
    I. Macdonald, Symmetric functions and Hall polynomials, 2nd ed., Oxford University Press, Oxford, 1995.Google Scholar
  16. 16.
    J. Maillet and F. Nijhoff, “Multidimensional latttice integrability and the simplex equations”, in Nonlinear Evolution Equations: Integrability and Spectral Methods, A. Degasperis, A. Fordy, and M. Lakshmanan, eds., Manchester University Press, 1990, 537-548.Google Scholar
  17. 17.
    Y. Manin and V. Schechtman, “Arrangements of real hyperplanes and Zamolodchikov equations”, in Group Theoretical Methods in Physics, M. Markov, V. Man’ko, and V. Dodonov, eds., Adv. Stud. Pure Math., vol. 17, VNU Science Press, 1986, 151-165.Google Scholar
  18. 18.
    _____, “Higher Bruhat orders, related to the symmetric group”, Funct. Anal. Appl. 20 (1986) 148-150.Google Scholar
  19. 19.
    _____, “Arrangements of hyperplanes, higher braid groups and higher Bruhat orders”, in Algebraic Number Theory - in honor ofK. Iwasawa, J. Coates, R. Greenberg, B. Mazur, and I. Satake, eds., Adv. Stud. Pure Math., vol. 17, Academic Press, 1989, 289-308.Google Scholar
  20. 20.
    F. Michielsen and F. Nijhoff, “D-algebras, the D-simplex equations, and multidimensional integrability”, in Quantum Topology, L. Kauffman and R. Baadhio, eds., Series on Knots and Everything, vol. 3, World Scientific, 1993, 230-243.Google Scholar
  21. 21.
    J. Perk and H. Au-Yang, “Yang-Baxter equations”, in Encyclopedia of Mathematical Physics, J.-P. Francoise, G. Naber, and S. Tsou, eds., vol. 5, Elsevier Science, 2006, 465-473.Google Scholar
  22. 22.
    J. Rambau and V. Reiner, “A survey of the higher Stasheff-Tamari orders”, in this volume. Google Scholar
  23. 23.
    R. Street, “The algebra of oriented simplexes”, J. Pure Appl. Alg. 49 (1987) 283-335.MathSciNetzbMATHCrossRefGoogle Scholar
  24. 24.
    D. Tamari, “Monoides préordonnés et chaînes de Malcev”, Doctorat és-Sciences Mathématiques Thése de Mathématique, Paris, 1951.Google Scholar
  25. 25.
    _____, “The algebra of bracketings and their enumeration”, Nieuw Arch. Wisk. 10 (1962) 131-146.Google Scholar
  26. 26.
    H. Thomas, “Maps between higher Bruhat orders and higher Stasheff-Tamari posets”, in FPSAC 15th Anniversary International Conference on Formal Power Series and Algebraic Combinatorics, K. Eriksson, A. Björner, and S. Linusson, eds., 2003.Google Scholar
  27. 27.
    M. Voevodski and M. Kapranov, “Free n-categories generated by a cube, oriented matroids, and higher Bruhat orders”, Funct. Anal. Appl. 25 (1990) 50-52.CrossRefGoogle Scholar
  28. 28.
    Wolfram Research, Inc., Mathematica 8.0, Wolfram Research, Inc., Champaign, Illinois, 2010.Google Scholar
  29. 29.
    A. Zamolodchikov, “Tetrahedron equations and the relativistic S-matrix of straight-strings in 2+1-dimensions”, Commun. Math. Phys. 79 (1981) 489-505.MathSciNetCrossRefGoogle Scholar
  30. 30.
    G. Ziegler, “Higher Bruhat orders and cyclic hyperplane arrangements”, Topology 32 (1993) 259-297.MathSciNetzbMATHCrossRefGoogle Scholar

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© Springer Basel 2012

Authors and Affiliations

  1. 1.Department of Financial and Management EngineeringUniversity of the AegeanChiosGreece
  2. 2.Max-Planck-Institute for Dynamics and Self-OrganizationGöttingenGermany

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