The Peregrine breather of order nine and its deformations with sixteen parameters solutions to the NLS equation

• We construct new deformations of the 9th Peregrine breather with 16 real parameters.
• We obtain explicitly new families of quasi-rational solutions to the NLS equation.
• When all the parameters are equal to 0, we recover the 9th Peregrine breather.
• We construct new patterns of rogue waves of 45 peaks as triangles or rings.

We construct new deformations of the Peregrine breather (P9P9) of order 9 with 16 real parameters. With this method, we obtain explicitly new families of quasi-rational solutions to the NLS equation in terms of a product of an exponential depending on t by a ratio of two polynomials of degree 90 in x and t  ; when all the parameters are equal to 0, we recover the classical P9P9 breather. We construct new patterns of different types of rogue waves as triangular configurations of 45 peaks as well as rings and concentric rings.