Abstract and Applied Analysis
Volume 2005 (2005), Issue 8, Pages 855-862

Existence and uniqueness of analytic solutions of the Shabat equation

Eugenia N. Petropoulou

Division of Applied Mathematics and Mechanics, Mechanics, Department of Engineering Sciences, School of Engineering, University of Patras, Patras 26500, Greece

Received 30 May 2004

Copyright © 2005 Eugenia N. Petropoulou. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


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:|z|<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.