PT-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 PT-symmetric potential. However, these solutions represent only the unbroken-PT phase owing to their isospectrality to an infinite potential well in the complex plane having real spectra. To obtain the broken-PT phase, an extension of the potential satisfying the sl2,R potential algebra is mandatory that additionally supports non-trivial zero-width resonances.