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<p>In this work, we propose second-order numerical integrators for solving the generalized Korteweg-de Vries (gKdV) equations, within the framework of gauge transformation, which is a framework to enhance the stability and accuracy of discretizations. Specifically, two numerical schemes have been introduced: the explicit gauged-transformed exponential integrator (EGTEI) and the time-symmetric gauged-transformed exponential integrator (SGTEI). Both schemes can be efficiently implemented using Fourier pseudo-spectral discretization in space. The rigorous convergence theorem for the gKdV equation in the classical regime is established, which highlights the optimal convergence order and the sharp regularity requirements. Numerical experiments show that EGTEI and SGTEI outperform traditional methods in terms of precision, particularly under rough data, and for the SGTEI scheme, the energy and Hamiltonian errors remain uniformly bounded in time. Applications to simulations of dispersive shock waves further illustrate the better efficiency and accuracy of the proposed methods.</p>

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Second Order Gauge-Transformed Exponential Integrators for gKdV Equations with Rough Data: Convergence and Application

  • 期刊论文

  • 发表日期：

    2026年1月17日

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  • Bing Li,
  • Qianrui Wei,
  • Xiaofei Zhao

摘要

In this work, we propose second-order numerical integrators for solving the generalized Korteweg-de Vries (gKdV) equations, within the framework of gauge transformation, which is a framework to enhance the stability and accuracy of discretizations. Specifically, two numerical schemes have been introduced: the explicit gauged-transformed exponential integrator (EGTEI) and the time-symmetric gauged-transformed exponential integrator (SGTEI). Both schemes can be efficiently implemented using Fourier pseudo-spectral discretization in space. The rigorous convergence theorem for the gKdV equation in the classical regime is established, which highlights the optimal convergence order and the sharp regularity requirements. Numerical experiments show that EGTEI and SGTEI outperform traditional methods in terms of precision, particularly under rough data, and for the SGTEI scheme, the energy and Hamiltonian errors remain uniformly bounded in time. Applications to simulations of dispersive shock waves further illustrate the better efficiency and accuracy of the proposed methods.