Explicit Fourier Integrator for the Periodic dNLS via Gauge Transformation: Low-Regularity Estimates in Discrete Bourgain Spaces
This work provides rigorous low-regularity error estimates for a numerical scheme for the periodic dNLS, a challenging problem due to lack of local smoothing and strong resonances.
The authors develop a filtered explicit Fourier integrator for the periodic derivative nonlinear Schrödinger equation using a gauge transformation, and prove convergence rates in $H^{1/2}$ for initial data in $H^s$ with $s>1/2$, achieving error $\mathcal{O}( au^{s/2-1/4})$. Numerical experiments confirm the predicted rates.
The derivative nonlinear Schrödinger equation is a fundamental model for the propagation of nonlinear dispersive waves in, for example, plasma physics and nonlinear optics. In this work, we consider this model on the one-dimensional torus and study a filtered explicit Fourier integrator for the corresponding periodic problem. After applying a periodic gauge transformation, we consider a frequency-truncated model and its filtered exponential-Euler discretization. The main difficulty comes from the derivative cubic nonlinearity in the periodic setting, since local smoothing is unavailable and resonant interactions are stronger than in the non-periodic case. To address this issue, we develop a discrete Bourgain-space framework adapted to the gauge-transformed equation. For initial data $u_0 \in H^s(\mathbb{T})$ with $1/2 < s \le 5/2$, we prove that the numerical error is of order $\mathcal{O}(τ^{s/2-1/4})$ in $H^{1/2}(\mathbb{T})$, where $τ$ denotes the employed time step size. Numerical experiments confirm the predicted convergence behavior and demonstrate the effectiveness of the filtered scheme for rough solutions.