Toward a classification of quasirational solutions of the nonlinear Schrödinger equation

Based on a representation in terms of determinants of the order $2N$, we attempt to classify quasirational solutions of the one-dimensional focusing nonlinear Schrödinger equation and also formulate several conjectures about the structure of the solutions. The se solutions can be written as a product of a $t$-dependent exponential times a quotient of two $N(N{+}1)$th degree polynomials in $x$ and $t$ depending on $2N{-}2$ parameters. It is remarkable that if all parameters are equal to zero in this representation, then we recover the $P_N$ breathers.