Abstract

We explore the Whitham modulation theory and one of its physical applications, the dam-breaking problem for the defocusing Hirota equation that describes the propagation of ultrashort pulses in optical fibers with third-order dispersion and self-steepening higher-order effects. By using the finite-gap integration approach, we deduce periodic solutions of the equation and discuss the degeneration of genus-one periodic solution to a soliton solution. Furthermore, the corresponding Whitham equations based on Riemann invariants are obtained, which can be used to modulate the periodic solutions with step-like initial data. These Whitham equations with the weak dispersion limit are quasilinear hyperbolic equations and elucidate the averaged dynamics of the fast oscillations referred to as dispersive shocks, which occur in the solution of the defocusing Hirota equation. We analyze the case where both characteristic velocities in genus-zero Whitham equations are equal to zero and the values of two Riemann invariants are taken as the critical case. Then by varying these two values as step-like initial data, we study the rarefaction wave and dispersive shock wave solutions of the Whitham equations. Under certain step-like initial data, the point where two genus-one dispersive shock waves begin to collide at a certain time, that is, the point where the genus-two dispersive shock wave appears, is investigated. We also discuss the dam-breaking problem as an important physical application of the Whitham modulation theory.

中文翻译：

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Whitham调制理论与散焦Hirota方程周期解下的溃坝问题

摘要

我们探索 Whitham 调制理论及其物理应用之一，散焦 Hirota 方程的破坝问题，该方程描述了具有三阶色散和自陡峭高阶效应的超短脉冲在光纤中的传播。利用有限间隙积分方法，推导了方程的周期解，并讨论了属一周期解对孤子解的退化。此外，还得到了相应的基于黎曼不变量的Whitham方程，该方程可用于调制具有阶梯状初始数据的周期解。这些具有弱色散极限的 Whitham 方程是拟线性双曲方程，阐明了被称为色散激波的快速振荡的平均动力学，这种振荡发生在散焦 Hirota 方程的解中。我们分析了亏格零Whitham方程中两个特征速度都等于0并且取两个黎曼不变量的值作为临界情况。然后通过将这两个值改变为阶梯状初始数据，我们研究了 Whitham 方程的稀疏波和色散冲击波解。在一定的阶梯状初始数据下，考察在某一时刻两个属一色散激波开始碰撞的点，即属二色散激波出现的点。我们还讨论了溃坝问题作为惠瑟姆调制理论的一个重要物理应用。