It is shown that asymptotically consistent modifications of (Boussinesq-like) shallow water approximations, in order to improve their dispersive properties, can fail for uneven bottoms (i.e., the dispersion is actually not improved). It is also shown that these modifications can lead to ill-posed equations when the water depth is not constant. These drawbacks are illustrated with the (fully nonlinear, weakly dispersive) Serre equations. We also derive asymptotically consistent, well-posed, modified Serre equations with improved dispersive properties for constant slopes of the bottom.

中文翻译：

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关于底面不均匀的色散改进的浅水方程的说明

结果表明，（Boussinesq-like）浅水近似值的渐近一致修改，以改善其分散特性，对于不均匀的底部可能会失败（即，实际上色散没有得到改善）。还表明，当水深不恒定时，这些修改会导致方程不适定。这些缺点用（完全非线性，弱色散）Serre方程表示。我们还导出了渐近一致的，位置适当的，经过改进的Serre方程，这些方程对底部的恒定斜率具有改善的色散特性。