A study of shallow-water waves

研究非线性浅水波数学模型中波浪破碎和峰值孤立波的稳定性问题，特别是Camassa-Holm （CH）方程、Degasperis-Procesi （DP）方程和双组分Camassa-Holm （CH2）系统。浅水波被定义为波长远远大于水深的波。当可能以不同速度和不同方向运动的波相交时，就会发生非线性波的相互作用。这些方程和系统具有永久波和破波：波的剖面保持有界，但其斜率在有限时间内变得无界。CH和DP方程也包含峰值孤波，因为这些波形复制了最大可能振幅的控制方程的行波解的特征特征。该项目提供了对波浪破碎现象的更深入的理解，并有助于浅水波浪的广泛建模。水波是现代应用数学和理论物理的前沿。200多年来，水波现象的研究一直是数学理论的丰富来源，并影响着与海洋和环境有关的各种问题。破浪，无论是白浪还是海浪，在海洋中都很常见，但令人惊讶的是，人们对它们知之甚少。然而，由于种种原因，它们很重要。它们在人造结构上施加巨大的水动力载荷，将水平动量传递给表层水流，提供湍流能量来源来混合海洋上层，并移动浅水中的沉积物。研究者研究非线性浅水波的数学模型，特别是CH和DP方程以及CH2系统。ch型方程也可能与一种重要的浅水破波现象——海啸——的建模有关。当一大片水域（如湖泊或海洋中的一个区域）迅速发生大规模位移时，就会产生海啸波。海啸的典型波长为200公里，受浅水方程式控制，当它们到达陆地时可能是灾难性的，正如最近在印度尼西亚和日本地震中所看到的那样。该项目旨在进一步了解浅水波浪破碎的动力学，特别是在水流稳定的情况下，破碎发生前后，更适合于理论和数值研究。特别是，研究人员进行与海啸波建模相关的数学分析，这反过来有助于更好地预测和了解波浪的特征。

LiuDMS-1207840 The investigator studies problems of wave breaking and the stability of peaked solitary waves in mathematical models of nonlinear shallow-water waves, particularly the Camassa-Holm (CH) equation, the Degasperis-Procesi (DP) equation, and the two-component Camassa-Holm (CH2) systems. Shallow-water waves are defined as waves whose wavelengths are far greater than the water depth. Nonlinear wave interactions occur when waves, moving possibly with different speeds and in different directions, intersect. These equations and systems have permanent waves and breaking waves: the wave profile remains bounded, but its slope becomes unbounded in finite time. The CH and DP equations also contain peaked solitary waves, as these wave forms replicate a feature characteristic of the traveling wave solutions to the governing equations for largest possible amplitude. This project provides a greater understanding of wave breaking phenomena and contributes to the extensive modeling of shallow-water waves. Water waves lie at the forefront of modern applied mathematics and theoretical physics. The study of water wave phenomena has been a rich source of mathematical theories for over 200 years and affects a variety of issues pertaining to the ocean and environment. Breaking waves, both whitecaps and surf, are commonly observed in the ocean, but surprisingly, little is known about them. Yet they are important for a variety of reasons. They place large hydrodynamic loads on man-made structures, transfer horizontal momentum to surface currents, provide a source of turbulent energy to mix the upper layers of the ocean, and move sediment in shallow water. The investigator studies mathematical models of nonlinear shallow-water waves, particularly the CH and DP equations as well as the CH2 systems. The CH-type equations might also be relevant to the modeling of an important shallow-water breaking wave phenomenon, the tsunami. A tsunami wave is generated when a large body of water, such as a region in a lake or a sea, becomes rapidly displaced on a massive scale. With typical wavelengths of 200 km, tsunamis are governed by shallow water equations and can be catastrophic when they reach land, as seen in the recent Indonesia and Japan earthquakes. This project seeks further understanding of the dynamics of shallow-water wave breaking, especially before and after breaking has occurred when the flow is steady and more amenable to theoretical and numerical study. In particular, the investigator conducts mathematical analysis pertinent to the modeling of tsunami waves, which in turn helps better predict and understand the waves' characteristics.