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

We construct new deformations of the Peregrine breather (P-9) 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 P-9 breather. We construct new patterns of different types of rogue waves as triangular configurations of 45 peaks as well as rings and concentric rings. (C) 2015 Elsevier B.V. All rights reserved.