Tenth Peregrine breather solution to the NLS equation

In this paper we construct a particularly important solution to the focusing NLS equation, namely a Peregrine breather of the rank 10 which we call, P-10 breather. The related explicit formula is given by the ratio of two polynomials of degree 110 with integer co-efficients times trivial exponential factor. This formula drastically simplifies for the "initial values" namely for t = 0 or x = 0. This formula confirms a general conjecture saying that between all quasi-rational solutions of the rank N fixed by the condition that its absolute value tends to 1 at infinity and its highest maximum is located at the point (x = 0, t = 0), the P-N breather is distinguished by the fact that P-N (0, 0) = 2N + 1 and, in the aforementioned class of quasi-rational solutions, it is an absolute maximum. At the end we also make a few remarks concerning the rational deformations of P-10 breather involving 2N - 2 free real parameters chosen in a way that P-N breather itself corresponds to the zero values of these parameters although we have no intention to discuss the properties of these deformations here. (C) 2015 Elsevier Inc. All rights reserved.