Low-regularity exponential-type integrators for the Zakharov system with rough data in all dimensions

Abstract:

We propose and analyze a type of low-regularity exponential-type integrators (LREIs) for the Zakharov system (ZS) with rough solutions. Our LREIs include a first-order integrator and a second-order one, and they achieve optimal convergence under weaker regularity assumptions on the exact solution compared to the existing numerical methods in literature. Specifically, the first-order integrator exhibits linear convergence in $H^{m+2}(\mathbb {T}^d)\times H^{m+1}(\mathbb {T}^d)\times H^m(\mathbb {T}^d)$ for solutions in $H^{m+3}(\mathbb {T}^d)\times H^{m+2}(\mathbb {T}^d)\times H^{m+1}(\mathbb {T}^d)$ if $m>d/2$, meaning that only the boundedness of one additional derivative of the solution is required to achieve the first-order convergence. While for the second-order integrator, we show that it achieves second-order accuracy by requiring the boundedness of two additional spatial derivatives of the solution. The order of additional derivatives required is reduced by half compared to the classical trigonometric integrators. The main techniques to design the integrators include a reformulation by introducing new variables to exclude the loss of spatial regularity in the original ZS, accurate integration for the dominant term in the linear part of the equations and appropriate approximations (or averaging approximations) to the exponential phase functions involving the nonlinear interactions. Numerical comparisons with classical integrators confirm that our newly proposed LREIs are superior in accuracy and robustness for handling rough data.