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A novel class of solutions of the non-stationary Schrödinger and the Kadomtsev-Petviashvili I equations. (English) Zbl 0947.35129

Summary: A new class of real, non-singular and rationally decaying potentials and eigenfunctions of the non-stationary Schrödinger equation and solutions of the KP I equation are constructed via binary Darboux transformations. These solutions are classified by the pole structure of the corresponding meromorphic eigenfunction and a set of integers including a quantity called the charge. The properties of the potential, eigenfunction and their relationship to the inverse scattering transform are discussed.

MSC:

35Q53 KdV equations (Korteweg-de Vries equations)
37K35 Lie-Bäcklund and other transformations for infinite-dimensional Hamiltonian and Lagrangian systems
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[1] M.J. Ablowitz, P.A. Clarkson, Solitons, Nonlinear Evolution Equations and Inverse Scattering, Cambridge Univ. Press, Cambridge, 1991.; M.J. Ablowitz, P.A. Clarkson, Solitons, Nonlinear Evolution Equations and Inverse Scattering, Cambridge Univ. Press, Cambridge, 1991. · Zbl 0762.35001
[2] Ablowitz, M. J.; Villarroel, J., Phys. Rev. Lett., 78, 570 (1997)
[3] J. Villarroel, M.J. Ablowitz, to appear in Commun. Math. Phys., 1999.; J. Villarroel, M.J. Ablowitz, to appear in Commun. Math. Phys., 1999.
[4] Fokas, A. S.; Ablowitz, M. J., Stud. Appl. Math., 69, 211 (1983)
[5] V.B. Matveev, M.A. Salle, Darboux Transformations and Solitons, Springer, Berlin, 1991.; V.B. Matveev, M.A. Salle, Darboux Transformations and Solitons, Springer, Berlin, 1991. · Zbl 0744.35045
[6] Johnson, R. S.; Thompson, S., Phys. Lett. A, 66, 279 (1978)
[7] Pelinovskii, D. E.; Stepanyants, Yu. A., JETP Lett., 57, 24 (1993)
[8] Galkin, V. M.; Pelinovskii, D. E.; Stepanyants, Yu. A., Phys. D, 80, 246 (1995)
[9] Pelinovskii, D. E., J. Math. Phys., 39, 5377 (1998)
[10] Liu, Q. P.; Manas, M., J. Nonlin. Sci., 9, 213 (1999)
[11] Steinberg, S., Arch. Rat. Mech. Anal., 31, 372 (1969)
[12] Segur, H., AIP Conf. Proc., 88, 211 (1982)
[13] Zhou, X., Commun. Math. Phys., 128, 551 (1990)
[14] Boiti, M.; Pempinelli, F.; Pogrebkov, A., Inv. Probl., 13, L7 (1997)
[15] Manakov, S. V.; Zakharov, V. E.; Bordag, L. A.; Its, A. R.; Matveev, V. B., Phys. Lett. A, 63, 205 (1977)
[16] A.D. Trubatch, Ph.D. Thesis, University of Colorado, Boulder, 1999.; A.D. Trubatch, Ph.D. Thesis, University of Colorado, Boulder, 1999.
[17] F.R. Gantmacher, The Theory of Matrices, Chelsea Publishing Company, New York, 1959.; F.R. Gantmacher, The Theory of Matrices, Chelsea Publishing Company, New York, 1959. · Zbl 0085.01001
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