Quasi-continuum modeling of the dispersion crossover in DNLS dam breaks

Develop a quasi-continuum asymptotic reduction for the defocusing discrete nonlinear Schrödinger equation i·u̇_n = −β(u_{n+1} + u_{n−1} − 2u_n) + |u_n|^2 u_n that accurately captures the “crossover” in dispersive shock wave morphology from negative to positive dispersion curvature as the coupling parameter β increases, for dam break initial data with small jump height |1 − u_+| ≪ 1. The model should reproduce the transition that the Kawahara reduction fails to describe when departing from its asymptotic validity, specifically in the weakly nonlinear long-wave regime of DNLS Riemann problems.

Background

In the weakly nonlinear long-wave regime, the authors derive and compare quasi-continuum reductions (KdV, KdV5, and Kawahara) against DNLS simulations for dam break problems with small amplitude steps. These reductions capture different dispersion regimes depending on the sign and magnitude of the third- and fifth-order dispersion coefficients, and provide good agreement in certain parameter ranges.

However, as β increases, the system quickly departs from the Kawahara regime where third- and fifth-order dispersion balance, and the reduction ceases to provide quantitatively accurate predictions—particularly for the transition to positive dispersion. The authors note that a better quantitative quasi-continuum approximation is needed to capture this crossover and suggest that full-dispersion Whitham models or Padé approximants may be promising approaches.