We consider the nonlinear classical field theory which results from adding to the Maxwell’s Lagrangian the contributions from the weak-field Euler–Heisenberg Lagrangian and a nonuniform part which involves derivatives of the electric and magnetic fields. We focus on the electrostatic case where the magnetic field is set to zero, and we derive the modified Gauss law, resulting in a higher-order differential equation. This equation gives the electric field produced by stationary charges in the higher-order nonlinear electrodynamics. Specializing for the case of a point charge, we investigate the solutions of the modified Gauss law and calculate the correction to the Coulomb law.

中文翻译：

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非均匀欧拉-海森堡理论中的库仑定律

我们考虑非线性经典场理论，它是由弱场欧拉-海森堡拉格朗日的贡献和一个涉及电场和磁场导数的非均匀部分添加到麦克斯韦拉格朗日的。我们专注于磁场设置为零的静电情况，并推导出修改后的高斯定律，从而得到一个高阶微分方程。该方程给出了高阶非线性电动力学中由静止电荷产生的电场。专门针对点电荷的情况，我们研究修改后的高斯定律的解并计算对库仑定律的修正。