Semina: Ciências Exatas e Tecnológicas, Londrina, v. 31, jan./jun. 2010 1 Solitons in Ideal Optical Fibers – A Numerical Development Sólitons em Fibras Óticas Ideais – Um Desenvolvimento Numérico Eliandro Rodrigues Cirilo 1 ; Paulo Laerte Natti 2 ; Neyva Maria Lopes Romeiro 3 ; Érica Regina Takano Natti 4 ; Camila Fogaça de Oliveira 5 1 Professor in the Mathematics Departament at Universidade Estadual de Londrina; ercirilo@uel.br 2 Professor in the Mathematics Departament at Universidade Estadual de Londrina; plnatti@uel.br 3 Professor in the Mathematics Departament at Universidade Estadual de Londrina; nromeiro@uel.br 4 Professor in the Pontifícia Universidade Católica do Paraná - Londrina; erica.natti@pucpr.br 5 Graduated in Mathematics from Universidade Estadual de Londrina; ca_fogaca@yahoo.com.br Abstract This work developed a numerical procedure for a system of partial differential equations (PDEs) describing the propagation of solitons in ideal optical fibers. The validation of the procedure was implemented from the numerical comparison between the known analytical solutions of the PDEs system and those obtained by using the numerical procedure developed. It was discovered that the procedure, based on the finite difference method and relaxation Gauss-Seidel method, was adequate in describing the propagation of soliton waves in ideals optical fibers. Key words: Optical communication. Solitons. Finite differences. Relaxation Gauss-Seidel method. Resumo Este trabalho desenvolveu um procedimento numérico para um sistema de equações diferenciais parciais (EDP’s) que descreve a propagação de sólitons em fibras óticas ideais. A validação do procedimento foi implementada a partir da comparação numérica entre as soluções analíticas conhecidas do sistema de EDP’s e aquelas obtidas por meio do procedimento numérico desenvolvido. Verificou-se que o procedimento, baseado no método das diferenças finitas e no método de Gauss-Seidel com relaxação, mostrou-se adequado na descrição da propagação das ondas sólitons em fibras óticas ideais. Palavras-chave: Comunicação ótica. Sólitons. Diferenças finitas. Método de Gauss-Seidel com relaxação. 1 Introduction In the last decades, several experiments were carried out aiming at trying to improve the capacities of the optical communication systems. The important issue is how to compensate the dispersion and the nonlinearities in communication systems at long distances (thousands of kilometers) or in high debt ground systems. A good technique that allows simultaneous compensation of such effects had already been proposed by Hasegawa and Tappert, in 1973, though only after the appearance of the optical amplifier could it be applied to practical systems (HASEGAWA; TAPPERT, 1973). This technique is based on the use of optical pulses, whose electrical field has the shape of a hyperbolic secant with some milliwatts of peak potency, and in the compensation of the dispersion by the optical fiber nonlinearities. Such pulses, called solitons, are capable of self-propagation, keeping their shape unchanged in a dispersive and non-linear environment, like the optical fiber. (EILENBERGER, 1981; TAYLOR, 1992). In the 1980’s, the experimental development of communication systems based on optical solitons started. Mollenauer, Stolen and Gordon, in 1980, conducted the first experimental observation of the bright soliton propagation in optical fibers. Hasegawa, in 1984, proposed that optical solitons could be used in long distance communication without the need of repeating stations, including overseas communications (HASEGAWA, 1984; PILIPETSKII, 2006). Since then, several experiments were conducted with the objective of improving the transmission capacity of solitons in optical fibers. Emplit et al., in 1987, carried out the first experimental observation of the dark soliton propagation in optical fibers (EMPLIT et al., 1987). Mollenauer and Smith, in 1988, transmitted soliton pulses