Squared Eigenfunctions for the Sasa-Satsuma Equation

TL;DR

This paper derives squared eigenfunctions for the Sasa-Satsuma equation using a Riemann-Hilbert approach, providing a unified framework that clarifies their role in the linearized and adjoint equations of integrable systems.

Contribution

It introduces a general derivation of squared eigenfunctions for integrable equations with a $3\times 3$ spectral operator, linking them to variations of scattering data and potentials.

Findings

Squared eigenfunctions are sums of products of Jost functions and adjoint Jost functions.

The derivation applies to the Sasa-Satsuma equation and related hierarchy equations.

Appendix presents squared eigenfunctions for the Manakov system.

Abstract

Squared eigenfunctions are quadratic combinations of Jost functions and adjoint Jost functions which satisfy the linearized equation of an integrable equation. In this article, squared eigenfunctions are derived for the Sasa-Satsuma equation whose spectral operator is a $3\times 3$ system, while its linearized operator is a $2\times 2$ 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.