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Lane formation dynamics of driven 2D pair-ion plasmas is investigated in underdamped cases. Extensive Brownian dynamics simulation is performed to study the behavior of the system in the presence of both constant and time-varying external electric fields. Lanes are found to form when like particles move along or opposite to the applied field direction. The lane order parameter has been implemented to detect phase transition. For the constant external field case, investigations are performed at different field strengths, to analyze the phase transition from a disordered to a lane state. It is observed that in this case, the electric field strength must exceed a critical value above which lanes are formed distinctly. For the case of the oscillating electric field, the frequency of the external oscillating field is found to control the lane formation phenomenon. We show that if the frequency of the external field exceeds a critical value, the system exhibits a transition back to the disordered state. A simple method for calculating the critical field strength provides quantitative agreement between the calculated and simulated values of the critical field strength for the case of the constant external electric field. The calculated value of the critical frequency agrees qualitatively with our simulation results for the oscillating external electric field case. A comparative study with the overdamped case has been performed, which suggests that the critical field strength corresponding to the phase transition point is higher for the underdamped case as compared to the overdamped one.

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We introduce a paradigm for spatial and modal wave manipulation based on nonlinear phononic crystals and explore its potential for engineering wave control systems with tunable, adaptive, and multifunctional characteristics. Our approach exploits nonlinear mechanisms to stretch the frequency signature of the wave response and distribute it over multiple modes, thereby activating a mixture of modal characteristics and enabling functionalities associated with high-frequency optical modes, even while operating in the low-frequency regime. To elucidate the versatility of this approach, we consider different granular crystal configurations that span the available landscape of crystal topologies and wave control functionalities. The ability to switch between complementary functionalities allows rethinking nonlinear phononic crystals as programmable acoustic ports that form the building blocks of a new structural logic framework enabled by nonlinearity.