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Matrix spectral problem with multiple-order jumps and poles

Published online by Cambridge University Press:  17 February 2009

Zhuhan Jiang
Zhuhan Jiang
Affiliation:
Department of Mathematics Statistics and Computing Science, University of New England, Armidale NSW 2351, Australia.
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Abstract

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The inverse spectral method for a general N × N spectral problem for solving nonlinear evolution equations in one spacial and one temporal dimension is extended to include multi-boundary jumps and high-order poles and their explicit representations. It therefore provides a formalism to generate soliton solutions that correspond to higher-order poles of the spectral data.

Type
Research Article
Information
The ANZIAM Journal , Volume 37 , Issue 3 , January 1996 , pp. 320 - 333
Copyright
Copyright © Australian Mathematical Society 1996

References

[1] Ablowitz, M. J., Yaacov, D. Bar and Fokas, A. S., “On the inverse scattering transform of the Kadomtsev-Petiashvili equation”, Stud. Appl. Math. 69 (1983) 135–142.CrossRefGoogle Scholar
[2] Ablowitz, M. J. and Segur, H., Solitons and the inverse scattering transform (SIAM, Philadelphia 1981).CrossRefGoogle Scholar
[3] Beals, R. and Coifman, R. R., “Inverse scattering and evolution equations”, Comm. Pure Appl. Math. 38 (1985) 29–42.CrossRefGoogle Scholar
[4] Beals, R. and Coifman, R. R., “Scattering and inverse scattering for first order systems. II”, Inv. Prob. 3 (1987) 577–593.CrossRefGoogle Scholar
[5] Boiti, M., Leon, J. J. and Pempinelli, F., “A new spectral transform for the Davey-Stewartson I equation”, Phys. Lett. A 141 (1989) 101–107.CrossRefGoogle Scholar
[6] Bullough, R. K. and Caudrey, P. J. (eds.), Solitons, (Springer, Berlin, 1980).CrossRefGoogle Scholar
[7] Caudrey, P. J., “The inverse problem for a general N × N spectral equation”, D. Physica 6 (1982) 51–66.Google Scholar
[8] Caudrey, P. J., in: Soliton theory: a survey of results (Fordy, A. P. ed.) (Manchester Univ. Press, 1990).Google Scholar
[9] Fokas, A. S. and Ablowitz, M. J., “On the inverse scattering of the time-dependent Schrödinger equation and the associated Kadomtsev-Petviashvili (I) equation”, Stud. Appl. Math. 69, (1983) 211–228.CrossRefGoogle Scholar
[10] Fokas, A. S. and Santini, P. M., “Dromions and a boundary value problem for the the Davey Stewartson I equation”, Physica D 44 (1990) 99–130.CrossRefGoogle Scholar
[11] Jiang, Z., “Construction of scattering data for a class of multidimensional scattering operators”, Inv. Prob. 5 (1989) 349–374; “Non-isospectral problems related to DS and other 2 + 1 dimensional nonlinear evolution equations”, Inv. Prob. 9 (1993) Ll–8; “A matrix spectral problem in one complex space dimension”, J. Phys. A 26 (1993) L375–378.CrossRefGoogle Scholar
[12] Jiang, Z., “The ZK-ZNKS inverse scattering transform with poles of higher orders”, Phys. Lett. A 148 (1990) 57–62.CrossRefGoogle Scholar
[13] Z, Jiang, “The N × N inverse spectral problem with poles of multiple orders”, Physica Scripta 45 (1992) 65–68.Google Scholar
[14] Jiang, Z. and Bullough, R. K., “Combined and Riemann-Hilbert inverse methods for integrable nonlinear evolution equations in 2 + 1 dimensions”, J. Phys. A 20 (1987) L429.CrossRefGoogle Scholar
[15] Jiang, Z. and Rauch-Wojciechowski, S., “Integrable dymamical systems with quadratic and cubic nonlinearities related to graded Lie algebras”, J. Math. Phys. 32 (1991) 1720–1732.CrossRefGoogle Scholar
[16] Leon, J., “Nonlinear evolutions, spectral transform and solitons in 3 + 1 dimensions’, Phys. Lett. A 156 (1991) 277–285.CrossRefGoogle Scholar
[17] Sabatier, P. C., “Spectral transform for nonlinear evolutions in N dimensional space”, Preprint PM/91–11, Universite Montpellier II, Montpellier Cedex 05, France.Google Scholar
[18] Takahashi, M. and Konno, K., “N double pole solution for the modified Korteweg-de Vries equation by the Hirota's method”, J. Phys. Soc. Japan 58 (1989) 3505–3508.CrossRefGoogle Scholar
[19] Zhou, X., “Direct and inverse scattering transforms with arbitrary spectral singularities”, Commun. Pure Appl. Math. 42 (1989) 895–938.CrossRefGoogle Scholar