Nonlinear Wave Interactions

这项资助支持的工作将解决非线性波动动力学的问题，将使用分析和数值方法相结合进行研究。例如：（1）赤道大尺度大气波的非线性相互作用，它在全球天气中起着基础性的作用。通过地形和对流特性的波的非线性相互作用的新机制（例如：位涡、温度和湿度）进行研究。(II)非线性双曲波与介质非均匀性的相互作用。这些波在传播过程中发生扭曲，导致形成强烈耗散的冲击。过去的研究表明，不均匀性或与其他波的相互作用可以阻止这一过程。例如，气体动力学欧拉方程的一大类新的解决方案已被发现并将被研究。这些解决方案是吸引的时间演变，并可以实现大的压力变化，而不会打破和形成冲击。它们的存在引起了有趣的数学问题。更好地理解这种现象可以提供控制（或至少减少）某些气流中冲击影响的方法。(III)弱震的聚焦、反射和折射。强声波和弱冲击是普遍存在的。深入了解它们的行为具有重要的科学意义，并将有助于揭示许多重要的实际问题。在这方面还存在许多令人困惑的问题。例如，在许多情况下，实验表明三个弱激波在一个几乎平坦的位置上连接在一起，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预测弱激波在适当的条件下经过聚焦、反射或折射时振幅的无穷大。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，已经取得了许多进展，但仍然存在许多悬而未决的问题，无论是关于应用还是所使用的数学。该奖项将支持全球天气中波浪状现象的数学模型研究，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建模。

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.