| We construct a multi-parametric family of solutions of the focusing NLS equation from the known result describing the multi phase almost-periodic elementary solutions given in terms of Riemann theta functions. We give a new representation of their solutions in terms of Wronskians determinants of order 2N composed of elementary trigonometric functions. When we perform a special passage to the limit when all the periods tend to infinity, we get a family of quasi-rational solutions. This leads to efficient representations for the Peregrine breathers of orders N=1,, 2, 3, first constructed by Akhmediev and his co-workers and also allows to get a simpler derivation of the generic formulas corresponding the 3 or 6 rogue-waves formation in frame of the NLS model first explained by Matveev et al. in 2010. Our formulation allows to isolate easier the second or third order Peregrine breather from a "generic" solutions, and also to compute the Peregrine breathers of order 2 and 3 easier with respect to other approaches. In the cases N=2, 3 we get the comfortable formulas to study the deformation of higher Peregrine breather of order 2 to the 3 rogue-waves or order 3 to the 6 rogue-waves solution via variation of the free parameters of our construction. |
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