Ne - Chapter 19 Nonlinear Systems Nonlinearity is ubiquitous in physical phenomena Fluid and plasma mechanics gas dynamics elasticity relativity

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Chapter 19 Nonlinear Systems Nonlinearity is ubiquitous in physical phenomena. Fluid and plasma mechanics, gas dynamics, elasticity, relativity, chemical reactions, combustion, ecology, biomechanics, and many, many other phenomena are all governed by inherently nonlinear equations. (The one notable exception is quantum mechanics, which is a fundamentally linear theory. Recent attempts at grand uni±cation of all fundamental physical theories, such as string theory and conformal ±eld theory, [ 85 ], do venture into the nonlinear wilderness.) For this reason, an ever increasing proportion of modern mathematical research is devoted to the analysis of nonlinear systems. Why, then, have we devoted the overwhelming majority of this text to linear mathe- matics? The facile answer is that nonlinear systems are vastly more di²cult to analyze. In the nonlinear regime, many of the most basic questions remain unanswered: existence and uniqueness of solutions are not guaranteed; explicit formulae are di²cult to come by; linear superposition is no longer available; numerical approximations are not always su²ciently accurate; etc., etc. A more intelligent answer is that a thorough understanding of linear phenomena and linear mathematics is an essential prerequisite for progress in the nonlinear arena. Therefore, in this introductory text on applied mathematics, we have no choice but to ±rst develop the proper linear foundations in su²cient depth before we can realistically confront the untamed nonlinear wilderness. Moreover, many important physical systems are “weakly nonlinear”, in the sense that, while nonlinear e³ects do play an essential role, the linear terms tend to dominate the physics, and so, to a ±rst approximation, the system is essentially linear. As a result, such nonlinear phenomena are best understood as some form of perturbation of their linear approximations. The truly nonlinear regime is, even today, only sporadically modeled and even less well understood. The advent of powerful computers has fomented a veritable revolution in our under- standing of nonlinear mathematics. Indeed, many of the most important modern analytical techniques drew their inspiration from early computer forays into the uncharted nonlinear wilderness. However, despite dramatic advances in both hardware capabilities and so- phisticated mathematical algorithms, many nonlinear systems — for instance, Einsteinian gravitation — still remain beyond the capabilities of today’s computers. Space limitations restrict us to providing just a brief overview of some of the most important ideas, mathematical techniques, and new physical phenomena that arise when venturing into the nonlinear realm. In this chapter, we start with iteration of nonlinear functions. Building on our experience with iterative linear systems, as developed in Chap- ter 10, we will discover that functional iteration, when it converges, provides a powerful mechanism for solving equations and for optimization. On the other hand, even very 2/25/07 1037 c c 2006 Peter J. Olver
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This note was uploaded on 02/10/2012 for the course MATH 5485 taught by Professor Olver during the Fall '09 term at University of Central Florida.

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Ne - Chapter 19 Nonlinear Systems Nonlinearity is ubiquitous in physical phenomena Fluid and plasma mechanics gas dynamics elasticity relativity

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