中文摘要：

近20多年来,有关两类主要的水波深度平均方程（线性长波方程和缓坡类方程）的解析解研究取得了一系列进展。关于线性长波方程,对理想地形（水深函数为幂函数情形）和拟理想地形（水深函数为幂函数与一个常数之和的情形）构造了一系列准确解析解。其中,针对理想地形所构造的解析解一般为封闭解,而针对拟理想地形所构造的解析解一般只能写成Taylor级数或Frobenius级数的形式。关于缓坡类方程,于最近构造了一系列Taylor级数形式的准确解析解,解决了国际水波界40多年来的开问题。其中,针对分段单调且分段2阶光滑的2维地形以及分片单调且分片2阶光滑的轴对称3维地形,隐式的修正缓坡方程被成功转化为显式方程。本文对20多年来这两类深度平均水波方程解析求解的主要研究进展给予一个较全面系统的综述,并对该方面的研究前景做一些展望。

英文摘要：

In the past two decades, a series of exact analytical solutions to the two kinds of depth-averaged equations in water waves, i. e. , the linear long-wave equation and the mild-slope type equations, were constructed. For the linear long-wave equation, if the bottom topography is idealized with the water depth being a power function,then the related analytical solution can be written in a closed form. If the bottom topography is quasi-idealized with the water depth being a power function plus a constant,then the related analytical solu- tion can be expanded into a Taylor series or a Frobenius series. For the mild-slope type equations, a number of exact analytical solutions in the form of Taylor series were constructed recently, where the implicit modified mild-slope equation is successfully transformed into an explicit equation for both two-dimensional bathymetries and three-dimensional axisymmetric bathymetries with piecewise monotonicity and piecewise second-order smoothness. These advances were summarized and reviewed, and some prospects of the related research in the future were described.