Mathematical Sciences: Nonlinear Wave Motion

小行星9404265 本计画研究一类具物理意义的非线性波动方程的解。将考虑自对偶Yang-Mills系统的新约化。自对偶方程已被证明是一个主可积系统，几乎所有已知的孤子方程都包含在其中作为特例。新的解决方案类将进行研究，包括那些最近发现的，可以表示在经典的自守函数，其解决方案的结构具有复杂的奇异性，这是在复杂的平面上的自然边界。非线性可积系统将被用于计算模拟，以建立存在和分析理解的数值混沌现象。物理上重要的强迫非线性波动方程的新的解决方案将被开发。 我们所研究的许多非线性方程是物理现象中普遍重要的渐近近似。应用领域包括非线性光波传播、流体动力学中的内波和表面波、晶格动力学和相对论。孤子在远距离通信技术中的应用已受到广泛关注。在本研究计画中，我们将分析多孤子解及其相互作用的性质，以显著地改善以孤子为基础的通讯系统的效能。这通常被称为光通信中的波分复用（WDM）。将研究的强迫非线性波系统之一涉及表面波中的移动压力分布，例如以临界速度移动的船舶。

9404265 Ablowitz This project involves the study of solutions to a class of physically significant nonlinear wave equations. New reductions of the self dual Yang-Mills system will be considered. The self dual equations have been shown to be a master integrable system from which virtually all known soliton equations are contained as special cases. Novel classes of solutions will be studied including those recently found which can be expressed in terms of classical automorphic functions whose solution structure possess complex singularities which are natural boundaries in the complex plane. Nonlinear integrable systems will be used in computational simulations in order to establish the existence of and to analytically understand the phenomena of numerical chaos. New solutions to physically important forced nonlinear wave equations will be developed. Many of the nonlinear equations we are studying are universally important asymptotic approximations in physical phenomena. Applications include nonlinear optical wave propagation, internal and surface waves in fluid dynamics, lattice dynamics, and relativity. The use of solitons in the technological application of long distance communication has been widely noted. In this research project, multisoliton solutions and their interaction properties will be analyzed in order to significantly improve the effectiveness of soliton based communication systems. This is usually referred to as wave length division multiplexing (WDM) in optical communications. One of the forced nonlinear wave systems which will be studied involves moving pressure distributions in surface waves, such as ships moving at critical speeds.