$$\mathscr{P}\mathscr{T}$$-symmetric KdV solutions and their algebraic extension 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 $$\mathscr{P}\mathscr{T}$$ P T -symmetric potential. However, these solutions represent only the unbroken-$$\mathscr{P}\mathscr{T}$$ P T phase owing to their isospectrality to an infinite potential well in the complex plane having real spectra. To obtain the broken-$$\mathscr{P}\mathscr{T}$$ P T phase, an extension of the potential satisfying the $$sl\left( 2,\mathbb {R}\right)$$ s l 2 , R potential algebra is mandatory that additionally supports non-trivial zero-width resonances.