Periodic and Solitary Wave Solutions for the One-Dimensional Cubic Nonlinear Schrödinger Model

Abstract

Using a similar approach as Korteweg and de Vries, [19], we obtain periodic solutions expressed in terms of the Jacobi elliptic function cn, [3], for the self-focusing and defocusing one-dimensional cubic nonlinear Schrödinger equations. We will show that solitary wave solutions are recovered through a limiting process after the elliptic modulus of the Jacobi elliptic function cn that describes the periodic solutions for the self-focusing nonlinear Schrödinger model.

Keywords

• nls
• self-focusing
• defocusing
• dispersive
• nonlinearity
• carrier waves
• solution profile
• envelope
• cnoidal waves
• solitary waves
• surface gravity waves
• sound waves
• water-air interface
• sonic layer depth
• primary 33e05, 35q55
• secondary 35q53