Explicit higher order rational rogue waves of the nonlinear Schrödinger equation

The N-th order rational rogue wave of the nonlinear Schrödinger equation, which depends on 2N − 2 real parameters, has been shown to be impossible to generate by the nonlinear superposition formula. We here generate this sequence by a three-term recurrence relation, each step only requiring the computation of three N − 1th order determinants of 2N − 2 variables, while the previous method requires two determinants of order 2N in 2N variables. This allows us to obtain explicitly the seventh wave with its six arbitrary complex parameters. These very compact expressions open the possibility to investigate the possible existence of new patterns in addition to the already observed ones (concentric rings, polygonal configurations, …).