Methods and apparatus for implementing stable self-starting and self-sustaining electrical nonlinear pulse (e.g., soliton, cnoidal wave, or quasi-soliton) oscillators. In one example, a nonlinear pulse oscillator is implemented as a closed loop structure that comprises a nonlinear transmission line, an improved high-pass filter, and a nonlinear amplifier configured to provide a self-adjusting gain as a function of an average voltage of the oscillator signal, to provide a pulse waveform having a desired target amplitude. In one implementation, the nonlinear amplifier and high pass filter functions are integrated in a two stage nonlinear amplifier/filter apparatus employing complimentary NMOS and PMOS amplification components and associated filtering and feedback circuitry configured to essentially implement an electric circuit analog of a saturable absorber via an adaptive bias control technique.