\(\mathscr{P}\mathscr{T}\)-symmetric KdV solutions and their algebraic extension with zero-width resonances

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