Inverse scattering transform for the Tzitz\'{e}ica equation

TL;DR

This paper extends the inverse scattering transform to the Tzitzéica equation, introducing new eigenfunctions and symmetries, and constructs a Riemann-Hilbert problem to find exact solutions.

Contribution

It develops a novel inverse scattering framework for the Tzitzéica equation, including auxiliary eigenfunctions and a Riemann-Hilbert problem formulation.

Findings

Derived asymptotic behaviors of Jost eigenfunctions.

Constructed a Riemann-Hilbert problem for the inverse scattering.

Obtained new exact solutions for reflectionless potentials.

Abstract

The inverse scattering transform is extended to investigate the Tzitz\'{e}ica equation. A set of sectionally analytic eigenfunctions and auxiliary eigenfunctions are introduced. We note that in this procedure, the auxiliary eigenfunctions play an important role. Besides, the symmetries of the analytic eigenfunctions and scattering data are discussed. The asymptotic behaviors of the Jost eigenfunctions are derived systematically. A Riemann-Hilbert problem is constructed to study the inverse scattering problem. Lastly, some novel exact solutions are obtained for reflectionless potentials.