An electronic transmission line that contains an array of nonlinear elements (Josephson junctions) is studied theoretically. A continuous nonlinear wave equation describing the dynamics of the node flux along the transmission line is derived. It is shown that due to the nonlinearity of the system, a mixing process between four waves with different frequencies is possible. The mixing process can be utilized for amplification of weak signals due to the interaction with a strong pump wave. An analytical solution for the spatial evolution of the wave amplitudes is derived, and found to be in excellent agreement with the results of numerical computations. Simulations of realistic parameters show that the power gain can exceed 20 dB over a bandwidth of more than 2 GHz.

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Resonant three-wave interactions appear in many fields of physics e.g. nonlinear optics, plasma physics, acoustics and hydrodynamics. A general theory of autoresonant three-wave mixing in a nonuniform media is derived analytically and demonstrated numerically. It is shown that due to the medium nonuniformity, a stable phase-locked evolution is automatically established. For a weak nonuniformity, the efficiency of the energy conversion between the interacting waves can reach almost 100%. One of the potential applications of our theory is the design of highly-efficient optical parametric amplifiers.

New solutions to the coupled three-wave equations in a nonuniform plasma medium are presented that include both space and time dependence of the waves. By including the dominant nonlinear frequency shift of the material wave, it is shown that if the driving waves are sufficiently strong (in relation to the medium gradient), a nonlinearly phase-locked solution develops that is characteristic of autoresonance. In this case, the material (electrostatic) wave develops into a front starting at the linear resonance point and moving with the wave group velocity in a manner such that the intensity increases linearly with the propagation distance. The form of the other two (electromagnetic) waves follow naturally from the Manley-Rowe relations.

A theory of autoresonant four-wave mixing in tapered fibers is developed in application to optical parametric amplification (OPA). In autoresonance, the interacting waves (two pump waves, a signal, and an idler) stay phase-locked continuously despite variation of system parameters (spatial tapering). This spatially extended phase-locking allows complete pump depletion in the system and uniform amplification spectrum in a wide frequency band. Different aspects of autoresonant OPA are described including the automatic initial phaselocking, conditions for autoresonant transition, stability, and spatial range of the autoresonant interaction.