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Revista Colombiana de Matemáticas

versão impressa ISSN 0034-7426

Rev.colomb.mat. vol.53 no.1 Bogotá jan./jun. 2019

 

Original articles

Existence of periodic standing wave solutions for a system describing pulse propagation in an optical fiber

FELIPE ALEXANDER PIPICANO1 

JUAN CARLOS MUÑOZ GRAJALES2 

1 Universidad del Valle, Cali, Colombia. Departamento de Matemáticas, Universidad del Valle Facultad de Ciencias Calle 13 Nro 100-00 Cali, Colombia e-mail: felipe.pipicano@correounivalle.edu.co

2 Universidad del Valle, Cali, Colombia. Departamento de Matemáticas, Universidad del Valle Facultad de Ciencias Calle 13 Nro 100-00 Cali, Colombia e-mail: juan.munoz@correounivalle.edu.co


ABSTRACT.

We establish existence of periodic standing waves for a model to describe the propagation of a light pulse inside an optical fiber taking into account the Kerr effect. To this end, we apply the Lyapunov Center Theo-rem taking advantage that the corresponding standing wave equations can be rewritten as a Hamiltonian system. Furthermore, some of these solutions are approximated by using a Newton-type iteration, combined with a collocation-spectral strategy to discretize the system of standing wave equations. Our numerical simulations are found to be in accordance with our analytical results.

Key words and phrases: Schrodinger equations; standing wave solutions; non-linear optics; spectral scheme

RESUMEN.

Establecemos existencia de soluciones estacionarias periódicas para un modelo que describe la propagación de un pulso de luz en el interior de una fibra óptica teniendo en cuenta el efecto Kerr. Para este fin, aplicamos el Teorema Central de Lyapunov tomando ventaja de que las correspondientes ecuaciones de onda estacionaria pueden escribirse como un sistema Hamiltoniano. Además, algunas de estas soluciones son aproximadas usando una iteración de tipo Newton, combinada con un estrategia colocación-espectral para discretizar el sistema de ecuaciones de onda estacionaria. Las simulaciones numéricas presentadas se encuentran de acuerdo con nuestros resultados analíticos.

Palabras y frases clave: Ecuaciones Schrodinger; soluciones de onda estacionaria; óptica no lineal; esquema espectral

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Acknowledgements

This research was supported partially by Colciencias and Universidad del Valle, Calle 13 No. 100-00, Cali-Colombia, under the research project FP44842-080- 2016.

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Received: May 2017; Accepted: January 2019

2010 Mathematics Subject Classification. 35Q55, 35C07, 78A60, 65N35.

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