Higher order Peregrine breathers solutions to the NLS equation

The solutions to the one dimensional focusing nonlinear Schrödinger equation (NLS) can be written as a product of an exponential depending on t by a quotient of two polynomials of degree N(N + 1) in x and t. These solutions depend on 2N − 2 parameters : when all these parameters are equal to 0, we obtain the famous Peregrine breathers which we call PN breathers. Between all quasi-rational solutions of the rank N fixed by the condition that its absolute value tends to 1 at infinity and its highest maximum is located at the point (x = 0, t = 0), the PN breather is distinguished by the fact that PN (0, 0) = 2N + 1. We construct Peregrine breathers of the rank N explicitly for N ≤ 11. We give figures of these PN breathers in the (x; t) plane; plots of the solutions PN(0; t), PN (x; 0), never given for 6 ≤ N ≤ 11 are constructed in this work. It is the first time that the Peregrine breather of order 11 is explicitly constructed. PACS numbers : 33Q55, 37K10, 47.10A-, 47.35.Fg, 47.54.Bd