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Communications in Mathematical Physics

, Volume 208, Issue 3, pp 713–760 | Cite as

Eigenfunctions and Eigenvalues for a Scalar Riemann–Hilbert Problem Associated to Inverse Scattering

  • Dmitry E. Pelinovsky
  • Catherine Sulem

Abstract:

A complete set of eigenfunctions is introduced within the Riemann–Hilbert formalism for spectral problems associated to some solvable nonlinear evolution equations. In particular, we consider the time-independent and time-dependent Schrödinger problems which are related to the KdV and KPI equations possessing solitons and lumps, respectively. Non-standard scalar products, orthogonality and completeness relations are derived for these problems. The complete set of eigenfunctions is used for perturbation theory and bifurcation analysis of eigenvalues supported by the potentials under perturbations. We classify two different types of bifurcations of new eigenvalues and analyze their characteristic features. One type corresponds to thresholdless generation of solitons in the KdV equation, while the other predicts a threshold for generation of lumps in the KPI equation.

Keywords

Soliton Perturbation Theory Characteristic Feature Evolution Equation Scalar Product 
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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Copyright information

© Springer-Verlag Berlin Heidelberg 2000

Authors and Affiliations

  • Dmitry E. Pelinovsky
    • 1
  • Catherine Sulem
    • 1
  1. 1.Department of Mathematics, University of Toronto, Toronto, Ontario, M5S 3G3, CanadaCA

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