Multi-parametric Deformations of Peregrine Breathers Solutions to the NLS Equation

Aims/ The structure of the solutions to the one dimensional focusing nonlinear Schr¨odinger equation (NLS) for the order N in terms of quasi rational functions is given here. We first give the proof that the solutions can be expressed as a ratio of two wronskians of order 2N and then two determinants by an exponential depending on t with 2N − 2 parameters. It also is proved that for the order N, the solutions can be written as the product of an exponential depending on t by a quotient of two polynomials of degree N(N + 1) in x and t. The solutions depend on 2N − 2 parameters and give when all these parameters are equal to 0, the analogue of the famous Peregrine breather PN. It is fundamental to note that in this representation at order N, all these solutions can be seen as deformations with 2N − 2 parameters of the famous Peregrine breather PN. With this method, we already built Peregrine breathers until order N = 10, and their deformations depending on 2N − 2 parameters. We present here Peregrine breather of order 11 constructed for the first time.