On a direct approach to quasideterminant solutions of a noncommutative KP equation

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Gilson, C.R. ; Nimmo, J.J.C. (2007)

A noncommutative version of the KP equation and two families of its solutions expressed as quasideterminants are discussed. The origin of these solutions is explained by means of Darboux and binary Darboux transformations. Additionally, it is shown that these solutions may also be verified directly. This approach is reminiscent of the wronskian technique used for the Hirota bilinear form of the regular, commutative KP equation but, in the noncommutative case, no bilinearising transformation is available.
  • References (16)
    16 references, page 1 of 2

    [3] A. Dimakis and F. Mu¨ller-Hoissen. With a Cole-Hopf transformation to solutions of the noncommutative KP hierarchy in terms of Wronski matrices. J. Phys. A, 40(17):F321-F329, 2007.

    [4] P. Etingof, I. Gelfand, and V. Retakh. Factorization of differential operators, quasideterminants, and nonabelian Toda field equations. Math. Res. Lett., 4(2-3):413-425, 1997.

    [5] G. Fal′ki, F. Magri, M. Pedroni, and Kh. P. Subelli. An elementary approach to the polynomial τ -functions of the KP-hierarchy. Teoret. Mat. Fiz., 122(1):23-36, 2000.

    [6] I. M. Gelfand and L. A. Dickii. Fractional powers of operators and hamiltonian systems. Funct. Anal. Appl., (10):259-273, 1976.

    [7] I. M. Gelfand, S. Gelfand, V. S. Retakh, and R. L. Wilson. Quasideterminants. Adv. Math., 193(1):56-141, 2005.

    [8] I. M. Gelfand and V. S. Retakh. Determinants of matrices over noncommutative rings. Funktsional. Anal. i Prilozhen., 25(2):13-25, 96, 1991.

    [9] I. M. Gelfand and V. S. Retakh. Quasideterminants. I. Selecta Math. (N.S.), 3(4):517-546, 1997.

    [10] C. R. Gilson and J. J. C. Nimmo. On a direct approach to quasideterminant solutions of a noncommutative KP equation. J. Phys. A, 40(14):3839-3850, 2007.

    [11] V. M. Goncharenko and A. P. Veselov. Monodromy of the matrix Schr¨odinger equations and Darboux transformations. J. Phys. A, 31(23):5315-5326, 1998.

    [12] B.A. Kupershmidt. KP or mKP noncommutative mathematics of Lagrangian, Hamiltonian, and integrable systems, volume 78. American Mathematical Society, 2000.

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