Existence and uniqueness of analytic solutions of the Shabat equation

Abstract

Sufficient conditions are given so that the initial value problem for the Shabat equation has a unique analytic solution, which, together with its first derivative, converges absolutely for $z\in ℂ:\left|z\right|<T$, $T>0$. Moreover, a bound of this solution is given. The sufficient conditions involve only the initial condition, the parameters of the equation, and $T$. Furthermore, from these conditions, one can obtain an upper bound for $T$. Our results are in consistence with some recently found results.