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Squared eigenfunctions for the Sasa–Satsuma equation

### Abstract

Squared eigenfunctions are quadratic combinations of Jost functions and adjoint Jost functions which satisfy the linearized equation of an integrable equation. They are needed for various studies related to integrable equations, such as the development of its solitonperturbation theory. In this article, squared eigenfunctions are derived for the Sasa–Satsuma equation whose spectral operator is a system, while its linearized operator is a system. It is shown that these squared eigenfunctions are sums of two terms, where each term is a product of a Jost function and an adjoint Jost function. The procedure of this derivation consists of two steps: First is to calculate the variations of the potentials via variations of the scattering data by the Riemann–Hilbert method. The second one is to calculate the variations of the scattering data via the variations of the potentials through elementary calculations. While this procedure has been used before on other integrable equations, it is shown here, for the first time, that for a general integrable equation, the functions appearing in these variation relations are precisely the squared eigenfunctions and adjoint squared eigenfunctions satisfying, respectively, the linearized equation and the adjoint linearized equation of the integrable system. This proof clarifies this procedure and provides a unified explanation for previous results of squared eigenfunctions on individual integrable equations. This procedure uses primarily the spectral operator of the Lax pair. Thus two equations in the same integrable hierarchy will share the same squared eigenfunctions (except for a time-dependent factor). In the Appendix, the squared eigenfunctions are presented for the Manakov equations whose spectral operator is closely related to that of the Sasa–Satsuma equation.

© 2009 American Institute of Physics

Received 05 September 2008
Accepted 29 December 2008
Published online 05 February 2009

Acknowledgments:
We thank Dr. Taras Lakoba for very helpful discussions. The work of J.Y. was supported in part by the Air Force Office of Scientific Research under Grant No. USAF 9550-05-1-0379. The work of D.J.K. has been supported in part by the U.S. National Science Foundation under Grant No. DMS-0505566.

Article outline:

I. INTRODUCTION
II. THE RIEMANN–HILBERT PROBLEM
III. SQUARED EIGENFUNCTIONS AND THEIR CLOSURE RELATION
A. Variation of the potential and squared eigenfunctions
B. Variations of the scattering data and adjoint squared eigenfunctions
C. Extension to the general case
IV. SUMMARY AND DISCUSSION

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2009-02-05

2016-10-21

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