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D.E. Pelinovsky and C. Sulem

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Eigenfunctions and eigenvalues for a scalar
Riemann-Hilbert problem associated to inverse scattering

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Comm. Math. Phys. 208, 713-760 (2000)

**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
Schrodinger 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:**

TIME-DEPENDENT SCHRODINGER, BENJAMIN-ONO-EQUATION,
KADOMTSEV-PETVIASHVILI EQUATION, DAVEY-STEWARTSON,
SPECTRAL TRANSFORM, INTEGRABLE SYSTEMS