Mathematical Sciences: Nonlinear Wave Motion

9404265 Ablwitz这个项目涉及一类具有物理意义的非线性波动方程的解的研究。将考虑对Self Dual Yang-Mills系统进行新的削减。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。我们将研究新的解类，包括最近发现的那些可以用经典的自同构函数来表示的解，其解的结构具有复奇性，即复平面中的自然边界。在计算模拟中将使用非线性可积系统，以确定数值混沌的存在并分析地理解数值混沌现象。物理上重要的强迫非线性波动方程的新解将被开发出来。我们正在研究的许多非线性方程都是物理现象中普遍重要的渐近逼近。应用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。孤子在长距离通信的技术应用中的应用已经引起了广泛的注意。在本研究项目中，将对多孤子解及其相互作用特性进行分析，以显著提高基于孤子的通信系统的有效性。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浪系统之一涉及表面波中的运动压力分布，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.