Multi-parametric solutions to the NLS equation
The structure of the solutions to the one dimensional focusing nonlin-ear Schr{\"o}dinger 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.