Breathers and Soliton Solutions for a Generalization of the Nonlinear Schrödinger Equation

A generalized nonlinear Schrödinger equation, which describes the propagation of the femtosecond pulse in single mode optical silica fiber, is analytically investigated. By virtue of the Darboux transformation method, some new soliton solutions are generated: the bright one-soliton solution on the zero background, the dark one-soliton solution on the continuous wave background, the Akhmediev breather which delineates the modulation instability process, and the breather evolving periodically along the straight line with a certain angle ofx-axis andt-axis. Those results might be useful in the study of the femtosecond pulse in single mode optical silica fiber.