Three-Wave Mixing (TWM) processes appear in many fields of physics e.g. nonlinear optics, plasma physics, acoustics and hydrodynamics. In TWM, the sum of the frequencies of two waves is equal to the frequency of the third wave (ω1 + ω2 = ω3 ). Recently, a general theory of autoresonant three-wave mixing in a nonuniform medium has been derived analytically and demonstrated numerically . It has been shown that due to the medium nonuniformity, a stable phase-locked evolution is automatically established. For a weak nonuniformity, the conversion efficiency between the interacting waves can reach almost 100% of the pump energy (see Fig. 1).
Discharges between two electrodes behave in the same manner as lightning, streaking through space to create electric arcs where only the start and end points are fixed. Arc discharges are routinely used for machining  and micromachining , and for assisting the milling process . They are also used for fuel ignition in combustion engines [4–6] and as a means to control the hydrodynamics of high-speed gases [5, 6]. In view of these an future applications, it will be of fundamental importance to devise schemes through which arcing can be fully controlled.
Non-diffractive Airy beams attracted a great of attentions recently [1, 2]. Their unique properties led to many applications, such as curved plasma generation, particle manipulations, surface plasmon routing, electron acceleration, etc . Since the first prediction and experimental demonstration of optical Airy beams, many efforts have been made to maintain or control their self-accelerating trajectories in both linear and nonlinear media. Unfortunately, nonlinearities tend to destroy the phase of Airy beams and thus to break down the acceleration, especially under the action of a self-focusing nonlinearity.
Airy pulses are self-accelerating optical wavepackets featured by a quasi-nondispersive intense peak . In linear optics, they are the key ingredients for the realization of optical bullets, while in a nonlinear environment, they tend to shed off-shooting solitons under the effect of a Kerr nonlinearity or can be exploited to control super-continuum generation in highly nonlinear fibers. Such Airy pulses are analogous to spatial Airy beams which have been studied extensively due to their intriguing phenomena and potential applications.