Novel solitary pulses for a variable-coefficient derivative nonlinear Schrödinger equation

A derivative nonlinear Schrödinger equation with variable coefficient is considered. Special exact solutions in the form of a solitary pulse are obtained by the Hirota bilinear transformation. The essential ingredients are the identification of a special chirp factor and the use of wavenumbers dependent on time or space. The inclusion of damping or gain is necessary. The pulse may then undergo broadening or compression. Special cases, namely, exponential and algebraic dispersion coefficients, are discussed in detail. The case of exponential dispersion also permits the existence of a 2-soliton. This provides a strong hint for special properties, and suggests that further tests for integrability need to be performed. Finally, preliminary results on other types of exact solutions, e.g. periodic wave patterns, are reported.