Nonlinear Wave Interactions

这笔赠款支持的工作将解决非线性波浪动力学的问题，将使用解析和数值方法相结合的方法进行研究。例如：(I)赤道大尺度大气波的非线性相互作用，它们在全球天气中起着基本作用。通过地形和对流性质(例如：位涡、温度和湿度)，波的非线性相互作用的新机制将被研究。(Ii)非线性双曲波与介质不均匀的相互作用。这些波在传播时会扭曲，导致激波的形成，激波具有很强的耗散性。过去的研究表明，不均匀或与其他波的相互作用可以阻止这一过程。例如，找到了一大类气体动力学欧拉方程的新解，并将对其进行研究。这些解在时间演化上很吸引人，可以在不破裂和形成激波的情况下实现较大的压力变化。它们的存在引发了有趣的数学问题。更好地了解这一现象可以提供控制(或至少减少)某些气体流动中冲击的影响的手段。(3)弱地震的聚焦、反射和折射。强烈的声波和微弱的冲击无处不在。更好地了解它们的行为具有相当大的科学意义，并将有助于揭示许多重要的实际问题。这一领域仍然存在许多令人费解的问题。例如，在许多情况下，实验表明三个弱激波在一个点上以近乎平坦的构型连接，但理论计算表明这是不可能的，导致与实验相矛盾。第二个相关的问题是，当弱冲击波在适当的条件下经过聚焦、反射或折射时，对其幅度的无限大的预测。波出现在许多领域-如声学、大气和海洋科学、气体动力学、光学、燃烧理论等--它们从根本上影响动力学。在过去的几十年里取得了许多进展，但仍然有许多悬而未决的问题，无论是关于应用程序还是所使用的数学。该奖项将支持对全球天气中类似波浪现象的数学模型、避免冲击波形成的气体动力学过程以及复杂冲击波相互作用的数学模型的研究。

The work supported under this grant will address problems in Nonlinear Wave Dynamics, which will be studied using a combination of analytic and numerical methods. For example: (I) Nonlinear interactions of large scale atmospheric waves in the Equator, which play a fundamental role in the global weather. A new mechanism for nonlinear interactions of the waves via topography and convected properties (e.g.: potential vorticity, temperature and humidity) will be investigated. (II) Interaction of nonlinear hyperbolic waves with media nonuniformities. These waves distort as they propagate, leading to the formation of shocks, which are strongly dissipative. Past research shows that nonuniformities or interactions with other waves can stop this process. For example, a large class of new solutions to the Euler equations of Gas Dynamics was found and will be studied. These solutions are attracting for the time evolution and can achieve large pressure variations without breaking and forming shocks. Their existence gives rise to interesting mathematical issues. A better understanding of this phenomenon could provide means to control (or at least diminish) the influence of shocks in some gas flows. (III) Focusing, reflection and refraction of weak shocks. Strong acoustical waves and weak shocks are ubiquitous. A better understanding of their behavior has considerable scientific significance and will shed light into many important practical problems. Many puzzling problems remain in this area. For example, in many situations experiments show three weak shock waves joining at a point in a nearly flat configuration, but theoretical calculations indicate that this is not possible, leading to a contradiction with the experiments. A second related problem occurs with the prediction of infinities in the amplitudes of weak shock waves as they undergo focusing, reflection or refraction under appropriate conditions.Waves occur in many fields --- such as Acoustics, Atmosphere and Ocean science, Gas Dynamics, Optics, Combustion Theory, etc. --- where they fundamentally affect the dynamics. Many advances have been made in the last few decades, but many open questions remain,both concerning the applications and the mathematics that is used. This award will support research on mathematical models for wave-like phenomena in the global weather, processes in gas dynamics that avoid the formation of shock waves, and the mathematical modeling of complex shock wave interaction.