We show that, contrary to popular belief, lower order dispersive regularization of hyperbolic systems does not exclude the development of the localized shock-like transition fronts. To guide the numerical search of such solutions, we generalize Rankine-Hugoniot relations to cover the case of higher order dispersive discontinuities and study their properties in an idealized case of a transition between two periodic wave trains with different wave lengths. We present evidence that smoothed stationary fronts of this type are numerically stable in the case when regularization is temporal and one of the adjacent states is homogeneous. In the zero dispersion limit such shock-like transition fronts, that are not traveling waves and apparently require for their description more complex anzats, evolve into traveling wave type jump discontinuities.

中文翻译：

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色散系统中静止的类激波过渡前沿

我们表明，与流行的看法相反，双曲线系统的低阶色散正则化并不排除局部冲击状过渡前沿的发展。为了指导此类解的数值搜索，我们将 Rankine-Hugoniot 关系概括为涵盖高阶色散不连续性的情况，并在具有不同波长的两个周期波列之间过渡的理想化情况下研究它们的性质。我们提供的证据表明，在正则化是时间性的且相邻状态之一是均匀的情况下，这种类型的平滑静止锋面在数值上是稳定的。在零色散极限，这种类似激波的过渡前沿，不是行波，显然需要更复杂的 anzats 来描述，演变成行波型跳跃不连续性。