Other 2N-2 parameters solutions of the NLS equation and 2N+1 highest amplitude of the modulus of the Nth order AP breather

In this paper, we construct new deformations of the Akhmediev-Peregrine (AP) breather of order N (or AP(N) breather) with 2N - 2 real parameters. Other families of quasirational solutions of the nonlinear Schrodinger (NLS) equation are obtained. We evaluate the highest amplitude of the modulus of the AP breather of order N; we give the proof that the highest amplitude of the AP(N) breather is equal to 2N + 1. We get new formulas for the solutions of the NLS equation, which are different from these already given in previous works. New solutions for the order 8 and their deformations according to the parameters are explicitly given. We simultaneously get triangular configurations and isolated rings. Moreover, the appearance for certain values of the parameters and of new configurations of concentric rings are underscored.