Reappraisal of Whitham's 1967 theory for wave-mean flow interaction in shallow water

Numerical bifurcation and stability for the capillary-gravity Whitham equation

We adopt a robust numerical continuation scheme to examine the global bifurcation of periodic traveling waves of the capillary-gravity Whitham equation, which combines the dispersion in the linear theory of capillary-gravity waves and a shallow water nonlinearity. We employ a highly accurate numerical method for space discretization and time stepping, to address orbital stability and instability for a rich variety of the solutions. Our findings can help classify capillary-gravity waves and understand their long-term dynamics.

The Cubic Vortical Whitham Equation

The cubic-vortical Whitham equation is a model for wave motion on a vertically sheared current of constant vorticity in a shallow inviscid fluid. It generalizes the classical Whitham equation by allowing constant vorticity and by adding a cubic nonlinear term. The inclusion of this extra nonlinear term allows the equation to admit periodic, traveling-wave solutions with larger amplitude than the Whitham equation. Increasing vorticity leads to solutions with larger amplitude as well. The stability of these solutions is examined numerically. All moderate- and large-amplitude solutions, regardles…

Whitham方程中的超谐波不稳定性与波浪破碎

The superharmonic instability and wave breaking in Whitham equations

该研究证明Whitham方程及其三次推广的大振幅周期行波解存在超谐波不稳定性，阈值位于Hamilton量和L2范数最大值处，并比较了二者对Euler方程不同不稳定性的波陡阈值逼近精度。

守恒律、一类新的群不变解及其在 Whitham Broer Kaup 模型中的应用

Conservation laws, a new class of group invariant solutions, and its applications for the Whitham Broer Kaup model

本研究基于 Lie 对称分析为 Whitham Broer Kaup (WBK) 系统构建了一类新的解析与数值解，揭示了物理参数对波结构的影响，应用于海啸波传播，并推导了完整的局部守恒律集。

Some exact solutions to the Lighthill Whitham Richards Payne traffic flow equations II: moderate congestion

We find a further class of exact solutions to the Lighthill Whitham Richards Payne (LWRP) traffic flow equations. As before, using two consecutive Lagrangian transformations, a linearization is achieved. Next, depending on the initial density, we either obtain exact formulae for the dependence of the car density and velocity on x, t, or else, failing that, the same result in a parametric representation. The calculation always involves two possible factorizations of a consistency condition. Both must be considered. In physical terms, the lineup usually separates into two offshoots at different …

Formation and evolution of roll waves in a shallow free surface flow of a power-law fluid down an inclined plane

A laminar flow of a thin layer of mud down an inclined plane under the action of gravity is considered. The instability of a film flow and the formation of finite amplitude waves are studied in the framework of both two-dimensional governing equations of a power-law fluid and its depth-averaged hyperbolic simplification. Conditions of roll waves existence for these models are formulated in terms of Whitham criterion. Numerical calculations of the free surface evolution and the roll waves development are performed. It is shown that the roll waves amplitude obtained by the 2D equations is slight…

Modeling shallow water waves

We review here the derivation of many of the most important models that appear in the literature (mainly in coastal oceanography) for the description of waves in shallow water. We show that these models can be obtained using various asymptotic expansions of the "turbulent" and non-hydrostatic terms that appear in the equations that result from the vertical integration of the free surface Euler equations. Among these models are the well-known nonlinear shallow water (NSW), Boussinesq and Serre-Green-Naghdi (SGN) equations for which we review several pending open problems. More recent models suc…

变分单孤立波的性质研究：基于保守扩展KdV方程的应用于浅水色散激波

A study of variational single solitary waves governed by the conservative-extended KdV equation with applications to shallow water dispersive shocks

本文通过变分方法研究了由保守扩展KdV方程支配的单孤立波解，其方法更简洁且适用于实际问题，通过与数值模拟对比验证了理论预测与数值结果的一致性。

Dispersive dam-break flow of a photon fluid

We investigate the temporal photonic analogue of the dam-break phenomenon for shallow water by exploiting a fiber optics setup. We clearly observe the decay of the step-like input (photonic dam) into a pair of oppositely propagating rarefaction wave and dispersive shock wave. Our results show evidence for a critical transition of the dispersive shock into a self-cavitating state. The detailed observation of the cavitating state dynamics allows for a fully quantitative test of the Whitham modulation theory applied to the universal defocusing nonlinear Schroedinger equation.

自由表面流中斜向相互作用的孤立波与波尾迹

Obliquely interacting solitary waves and wave wakes in free-surface flows

本文针对描述浅海表面与内波的Benney--Luke方程，结合Whitham调制理论推导孤子演化相关方程与解析公式，验证了其与KP方程的一致性及数值吻合度，还得到受迫BL方程远场近似下的圆弧形前驱波前解。