Explicit higher order rational rogue waves of the nonlinear Schrödinger equation

The $N$-th order rational rogue wave of the nonlinear Schr\"odinger equation (NLS), which depends on $2 N-2$ 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 $N-1$-th order determinants of $2 N-2$ variables, while the previous method requires two determinants of order $2 N$ in $2 N$ 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, \dots).