# 𝒫𝒯-symmetric KdV solutions and their algebraic extension with zero-width resonances

## TL;DR

This paper explores complex wave solutions to KdV and mKdV equations using PT-symmetric potentials and extends them to include zero-width resonances.

## Contribution

The paper introduces an algebraic extension of PT-symmetric potentials to support zero-width resonances in the broken PT phase.

## Key findings

- Complex breather and soliton solutions are identified for KdV and mKdV equations with PT-symmetric potentials.
- An extension of the potential is required to access the broken PT phase with zero-width resonances.

## Abstract

A class of complex breather and soliton solutions to both KdV and mKdV equations are identified with a Pöschl-Teller type \documentclass[12pt]{minimal}
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\begin{document}$$\mathscr{P}\mathscr{T}$$\end{document}PT-symmetric potential. However, these solutions represent only the unbroken-\documentclass[12pt]{minimal}
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\begin{document}$$\mathscr{P}\mathscr{T}$$\end{document}PT phase owing to their isospectrality to an infinite potential well in the complex plane having real spectra. To obtain the broken-\documentclass[12pt]{minimal}
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\begin{document}$$\mathscr{P}\mathscr{T}$$\end{document}PT phase, an extension of the potential satisfying the \documentclass[12pt]{minimal}
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\begin{document}$$sl\left( 2,\mathbb {R}\right)$$\end{document}sl2,R potential algebra is mandatory that additionally supports non-trivial zero-width resonances.