Numerical simulation of nonlinear dispersive quantization
Gong Chen Peter J. Olver
Discrete & Continuous Dynamical Systems - A 2014, 34(3): 991-1008 doi: 10.3934/dcds.2014.34.991
When posed on a periodic domain in one space variable, linear dispersive evolution equations with integral polynomial dispersion relations exhibit strikingly different behaviors depending upon whether the time is rational or irrational relative to the length of the interval, thus producing the Talbot effect of dispersive quantization and fractalization. The goal here is to show that these remarkable phenomena extend to nonlinear dispersive evolution equations. We will present numerical simulations, based on operator splitting methods, of the nonlinear Schrödinger and Korteweg--deVries equations with step function initial data and periodic boundary conditions. For the integrable nonlinear Schrödinger equation, our observations have been rigorously confirmed in a recent paper of Erdoǧan and Tzirakis, [10].
keywords: Talbot effect operator splitting scheme Korteweg--deVries equation nonlinear Schrödinger equation. fractal Dispersion quantized
Strichartz estimates for charge transfer models
Gong Chen
Discrete & Continuous Dynamical Systems - A 2017, 37(3): 1201-1226 doi: 10.3934/dcds.2017050

In this note, we prove Strichartz estimates for scattering states of scalar charge transfer models in $\mathbb{R}^{3}$. Following the idea of Strichartz estimates based on [3,10], we also show that the energy of the whole evolution is bounded independently of time without using the phase space method, as for example, in [5]. One can easily generalize our arguments to $\mathbb{R}^{n}$ for $n≥q3$. We also discuss the extension of these results to matrix charge transfer models in $\mathbb{R}^{3}$.

keywords: Strichartz estimates charge transfer model energy boundedness

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