Dispersive shock waves in periodic lattices

Area of Science:

• Nonlinear Physics
• Wave Phenomena
• Condensed Matter Physics

Background:

• Dispersive shock waves (DSWs) occur in nonlinear systems like optical lattices and superfluids.
• The nonlinear Schrödinger (NLS) equation with periodic potentials models these phenomena.
• Understanding DSWs is crucial for applications in optics and quantum fluids.

Purpose of the Study:

• To investigate the generation and dynamics of DSWs in a nonlinear Schrödinger equation with a periodic potential.
• To develop and validate a reduced discrete model for analyzing these wave phenomena.
• To explore the rich spectrum of discrete dispersive hydrodynamic behaviors.

Main Methods:

• Utilized the tight-binding approximation to reduce the continuous NLS model to a discrete NLS (DNLS) model.
• Employed Whitham modulation theory and long-wave quasicontinuum reductions for analysis.
• Compared the discrete model's predictions with the continuous model's phenomenology.

Main Results:

• The tight-binding approximation shows higher fidelity for deeper periodic potentials.
• The reduced DNLS model accurately captures dynamics at the potential minima.
• A rich spectrum of nonconvex, discrete dispersive hydrodynamic phenomena was uncovered.

Conclusions:

• The discrete nonlinear Schrödinger model provides an effective framework for studying DSWs in periodic potentials.
• This work bridges the gap between continuous and discrete models for wave dynamics.
• The findings offer insights into wave propagation in structured nonlinear media.