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Construction of Inverse Data in Multidimensions

  • Zhuhan Jiang
Conference paper
Part of the Research Reports in Physics book series (RESREPORTS)

Abstract

The Inverse Scattering Transforms (ISTs) are among the main topics of this 5th workshop on NEEDS. Many participants have made their contributions on this subject [1]. After the first appearance of a \(\bar{\partial }\) method [2,3] in multidimensions, there are now a number of literatures in this connection. Since the review on ISTs [1] by other participants are already very comprehensive, we will only note that a general construction of inverse data are now present [4] for operator Q≡ P(∂1,...,∂M) — X(u,∂,...,∂M), polynomial in all arguments, as long as the Green function has no singularities other than jumps. However, extra cares are needed for some particular equations concerning discrete scattering data [5,6]. In this note, we only use a simple but general enough case to exemplify the procedure [4] of constructing inverse (scattering) data for multidimensional scattering operators, since a most general approach would have to go beyond the allowed length.

Keywords

Integral Equation Green Function Simple Polis General Construction Inverse Scattering 
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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References

  1. [1]
    Boiti, Fokas, Leon, Manakov, Sabatier, Santini, Sung et al, among the participants, have reviewed comprehensively on multidimensional inverse scattering transforms. See their contributions and the references there in this Proceedings.Google Scholar
  2. [2].
    M.J. Ablowitz, D.Bar Yaacov and A.S. Fokas, Stud.Appl.Math. 69, 135 (1983).Google Scholar
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    A.S. Fokas and M.J. Ablowitz, J. Math. Phys. 25, 2494 (1984).MathSciNetADSMATHCrossRefGoogle Scholar
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    Z. Jiang, Inverse Problem 5, 349 (1989).Google Scholar
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    M. Boiti, J. JP. Leon, F. Pempinelli, A new spectral transform for the Davey-Stewartson I equation, Preprint PM/89–10, Montpelier, 1989.Google Scholar
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    A.S. Fokas and P.M. Santini, Solitons in multidimensions, Preprint INS/106, Clarkson University, 1988.Google Scholar
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    If analytic V(z) and V-1(z) diagonalise an NxN polynomial A(z) to A(z), then A, V and V-1 are all polynomials for at least N 5, due to the standard solution formulae for the characteristic polynomial of A(z).Google Scholar
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    P.J. Caudrey, Private communication.Google Scholar

Copyright information

© Springer-Verlag Berlin, Heidelberg 1990

Authors and Affiliations

  • Zhuhan Jiang
    • 1
  1. 1.Department of MathematicsUMISTManchesterUK

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