We propose a semi-analytic Stokes expansion ansatz for finite-depth standing water waves and devise a recursive algorithm to solve the system of differential equations governing the expansion coefficients. We implement the algorithm on a supercomputer using arbitrary-precision arithmetic. The Stokes expansion introduces hyperbolic trigonometric terms that require exponentiation of power series. We handle this efficiently using Bell polynomials. Under mild assumptions on the fluid depth, we prove that there are no exact resonances, though small divisors may occur. Sudden changes in growth rate in the expansion coefficients are found to correspond to imperfect bifurcations observed when families of standing waves are computed using a shooting method. A direct connection between small divisors in the recursive algorithm and imperfect bifurcations in the solution curves is observed, where the small divisor excites higher-frequency parasitic standing waves that oscillate on top of the main wave. A 109th order Pad\'e approximation maintains 25--30 digits of accuracy on both sides of the first imperfect bifurcation encountered for the unit-depth problem. This suggests that even if the Stokes expansion is divergent, there may be a closely related convergent sequence of rational approximations.

翻译：我们针对有限深度驻立水面波动提出一种半解析Stokes展开假设，并设计出递归算法以求解控制展开系数的微分方程组。我们在超级计算机上采用任意精度算术实现该算法。Stokes展开引入了需要幂级数指数运算的双曲三角项，我们利用Bell多项式高效处理这一问题。在关于流体深度的弱假设下，我们证明不存在精确共振，但可能出现小除数现象。展开系数增长率的突变与使用打靶法计算驻立波族时观察到的不完美分岔相对应。我们观察到递归算法中的小除数与解曲线中的不完美分岔之间存在直接联系——小除数会激发在主波上振荡的高频寄生驻立波。对于单位深度问题，第109阶Padé近似在首次遇到的不完美分岔两侧均保持25-30位有效数字精度。这表明即使Stokes展开发散，仍可能存在密切相关的收敛有理近似序列。