General rogue wave solutions to the discrete nonlinear Schr\"odinger equation

Rogue waves (RWs) or freak waves are spontaneously excited local nonlinear waves with large amplitudes which appear from nowhere and disappear with no trace [1]. The simplest form of such waves were firstly discovered Peregrine in the nonlinear Schrödinger (NLS) equation [2], and their higher order forms were found 20 years later in [3, 4, 5, 6, 7, 8, 9]. Such extreme wave have been observed in various different contexts such as oceanography [10], hydrodynamic [11, 12], Bose-Einstein condensate [13], plasma [14] and nonlinear optic [11, 15, 16]. Motivated by these physical applications, rogue wave solutions have been found in many other nonlinear wave equations such as the derivative Schrödinger (NLS) equation [17, 18, 19, 20, 21], the Manakov system [22, 23, 24], Davey-Stewartson I and II equation [25, 26], the three-wave equation [27], the Boussinesq equation [28], the Yajima-Oikawa equation [29, 30],.