Patterns of deformations of Peregrine breather of order 3 and 4 solutions to the NLS equation with multi parameters

In this article, one gives a classification of thesolutions to the one dimensional nonlinear focusingSchro¨dinger equation (NLS) by considering the modulus ofthe solutions in the (x, t) plan in the cases of orders 3 and 4.For this, we use a representation of solutions to NLSequation as a quotient of two determinants by an exponentialdepending on t. This formulation gives in the caseof the order 3 and 4, solutions with, respectively 4 and 6parameters. With this method, beside Peregrine breathers,we construct all characteristic patterns for the modulus ofsolutions, like triangular configurations, ring and others.