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

SIGMA 11 (2015), 087, 22 pages      arXiv:1505.02833      https://doi.org/10.3842/SIGMA.2015.087

Bispectrality of $N$-Component KP Wave Functions: A Study in Non-Commutativity

Alex Kasman
Department of Mathematics, College of Charleston, USA

Received May 13, 2015, in final form October 28, 2015; Published online November 01, 2015

Abstract
A wave function of the $N$-component KP Hierarchy with continuous flows determined by an invertible matrix $H$ is constructed from the choice of an $MN$-dimensional space of finitely-supported vector distributions. This wave function is shown to be an eigenfunction for a ring of matrix differential operators in $x$ having eigenvalues that are matrix functions of the spectral parameter $z$. If the space of distributions is invariant under left multiplication by $H$, then a matrix coefficient differential-translation operator in $z$ is shown to share this eigenfunction and have an eigenvalue that is a matrix function of $x$. This paper not only generates new examples of bispectral operators, it also explores the consequences of non-commutativity for techniques and objects used in previous investigations.

Key words: bispectrality; multi-component KP hierarchy; Darboux transformations; non-commutative solitons.

pdf (486 kb)   tex (32 kb)

References

  1. Bakalov B., Horozov E., Yakimov M., General methods for constructing bispectral operators, Phys. Lett. A 222 (1996), 59-66, q-alg/9605011.
  2. Bergvelt M., Gekhtman M., Kasman A., Spin Calogero particles and bispectral solutions of the matrix KP hierarchy, Math. Phys. Anal. Geom. 12 (2009), 181-200, arXiv:0806.2613.
  3. Bergvelt M.J., ten Kroode A.P.E., Partitions, vertex operator constructions and multi-component KP equations, Pacific J. Math. 171 (1995), 23-88, hep-th/9212087.
  4. Boyallian C., Liberati J.I., Matrix-valued bispectral operators and quasideterminants, J. Phys. A: Math. Theor. 41 (2008), 365209, 11 pages.
  5. Castro M.M., Grünbaum F.A., The algebra of differential operators associated to a family of matrix-valued orthogonal polynomials: five instructive examples, Int. Math. Res. Not. 2006 (2006), 47602, 33 pages.
  6. Chalub F.A.C.C., Zubelli J.P., Matrix bispectrality and Huygens' principle for Dirac operators, in Partial Differential Equations and Inverse Problems, Contemp. Math., Vol. 362, Amer. Math. Soc., Providence, RI, 2004, 89-112.
  7. Date E., Jimbo M., Kashiwara M., Miwa T., Transformation groups for soliton equations. III. Operator approach to the Kadomtsev-Petviashvili equation, J. Phys. Soc. Japan 50 (1981), 3806-3812.
  8. Duistermaat J.J., Grünbaum F.A., Differential equations in the spectral parameter, Comm. Math. Phys. 103 (1986), 177-240.
  9. Duran A.J., Matrix inner product having a matrix symmetric second order differential operator, Rocky Mountain J. Math. 27 (1997), 585-600.
  10. Etingof P., Gelfand I., Retakh V., Factorization of differential operators, quasideterminants, and nonabelian Toda field equations, Math. Res. Lett. 4 (1997), 413-425, q-alg/9701008.
  11. Fock V., Gorsky A., Nekrasov N., Rubtsov V., Duality in integrable systems and gauge theories, J. High Energy Phys. 2000 (2000), no. 7, 028, 40 pages, hep-th/9906235.
  12. Geiger J., Horozov E., Yakimov M., Noncommutative bispectral Darboux transformations, arXiv:1508.07879.
  13. Grünbaum F.A., Some noncommutative matrix algebras arising in the bispectral problem, SIGMA 10 (2014), 078, 9 pages, arXiv:1407.6458.
  14. Grünbaum F.A., Iliev P., A noncommutative version of the bispectral problem, J. Comput. Appl. Math. 161 (2003), 99-118.
  15. Grünbaum F.A., Pacharoni I., Tirao J., A matrix-valued solution to Bochner's problem, J. Phys. A: Math. Gen. 34 (2001), 10647-10656.
  16. Grünbaum F.A., Pacharoni I., Tirao J., Matrix valued spherical functions associated to the complex projective plane, J. Funct. Anal. 188 (2002), 350-441, math.RT/0108042.
  17. Grünbaum F.A., Pacharoni I., Tirao J., Matrix valued spherical functions associated to the three dimensional hyperbolic space, Internat. J. Math. 13 (2002), 727-784, math.RT/0203211.
  18. Grünbaum F.A., Pacharoni I., Tirao J., An invitation to matrix valued spherical functions: linearization of products in the case of the complex projective space $P_2(\mathbb{C})$, in Modern Signal Processing, MSRI Publications, Vol. 46, Editors D.N. Rockmore, D.M. Healy, Cambridge University Press, Cambridge, 2003, 147-160, math.RT/0202304.
  19. Haine L., KP trigonometric solitons and an adelic flag manifold, SIGMA 3 (2007), 015, 15 pages, nlin.SI/0701054.
  20. Harnad J., Kasman A. (Editors), The bispectral problem, CRM Proceedings & Lecture Notes, Vol. 14, Amer. Math. Soc., Providence, RI, 1998.
  21. Kasman A., Bispectral KP solutions and linearization of Calogero-Moser particle systems, Comm. Math. Phys. 172 (1995), 427-448, hep-th/9412124.
  22. Kasman A., Darboux transformations from $n$-KdV to KP, Acta Appl. Math. 49 (1997), 179-197.
  23. Kasman A., Spectral difference equations satisfied by KP soliton wavefunctions, Inverse Problems 14 (1998), 1481-1487, solv-int/9811009.
  24. Kasman A., Factorization of a matrix differential operator using functions in its kernel, arXiv:1509.05105.
  25. Kasman A., Rothstein M., Bispectral Darboux transformations: the generalized Airy case, Phys. D 102 (1997), 159-176, q-alg/9606018.
  26. Ruijsenaars S.N.M., Action-angle maps and scattering theory for some finite-dimensional integrable systems. I. The pure soliton case, Comm. Math. Phys. 115 (1988), 127-165.
  27. Sakhnovich A., Zubelli J.P., Bundle bispectrality for matrix differential equations, Integral Equations Operator Theory 41 (2001), 472-496.
  28. Sato M., Sato Y., Soliton equations as dynamical systems on infinite-dimensional Grassmann manifold, in Nonlinear Partial Differential Equations in Applied Science (Tokyo, 1982), North-Holland Math. Stud., Vol. 81, North-Holland, Amsterdam, 1983, 259-271.
  29. Segal G., Wilson G., Loop groups and equations of KdV type, Inst. Hautes Études Sci. Publ. Math. 61 (1985), 5-65.
  30. Wilson G., Bispectral commutative ordinary differential operators, J. Reine Angew. Math. 442 (1993), 177-204.
  31. Wilson G., Collisions of Calogero-Moser particles and an adelic Grassmannian, Invent. Math. 133 (1998), 1-41.
  32. Wilson G., Notes on the vector adelic Grassmannian, arXiv:1507.00693.
  33. Zubelli J.P., Differential equations in the spectral parameter for matrix differential operators, Phys. D 43 (1990), 269-287.

Previous article  Next article   Contents of Volume 11 (2015)