The periodic Cauchy problem for PT-symmetric NLS, I: the first appearance of rogue waves, regular behavior or blow up at finite times

The recently discovered PT-symmetric nonlinear Schrödinger (PT–NLS) equation is an integrable nonlinear and nonlocal dispersive model. In the focusing case, monochromatic perturbations of the constant background solution with sufficiently small wave number are modulationally unstable and one expects the formation of rogue waves (RWs), as for the celebrated focusing nonlinear Schrödinger (NLS) equation. In this paper we investigate the x-periodic Cauchy problem of PT–NLS, for a generic periodic initial perturbation of the constant background solution (what we call the periodic RW problem), in the simplest case of one unstable mode only. We use matched asymptotic expansion techniques to study the first appearance of RWs, well described by two recently discovered exact solutions of PT–NLS, whose free parameters can be expressed in terms of the initial data through elementary functions. Depending on the initial data, the rogue waves are either regular, with arbitrarily large amplitude, or they blow up twice at the first appearance, unlike the NLS case, in which the RWs are always regular and with fixed amplitude. A qualitative reason for it should be the gain-loss properties of the complex self-induced potential of PT–NLS, that could cause extra-focusing effects with respect to the NLS case. This paper is motivated by recent works of Grinevich and the author in which a similar approach, as well as the finite gap method, have been used to solve the RW periodic Cauchy problem for the focusing NLS equation.