Chapter 22 Nonlinear Partial Differential Equations

The ultimate topic to be touched on in this book is the vast and active field of nonlinear partial differential equations. Leaving aside quantum mechanics, which remains to date an inherently linear theory, most real-world physical systems, including gas dynamics, fluid mechanics, elasticity, relativity, ecology, neurology, thermodynamics, and many more, are modeled by nonlinear partial differential equations. Attempts to survey, in such a small space, even a tiny fraction of such an all-encompassing range of phenomena, methods, results, and mathematical developments, are doomed to failure. So we will be content to introduce a handful of prototypical, seminal examples that arise in the study of nonlinear waves and that serve to highlight some of the most significant physical and mathematical phenomena not encountered in simpler linear systems. We will only have space to look at simple one-dimensional models; the far more complicated nonlinear systems that govern our three-dimensional dynamical universe quickly lead one to the cutting edge of contemporary research. Historically, comparatively little was known about the extraordinary range of behavior exhibited by the solutions to nonlinear partial differential equations. Many of the most fundamental phenomena that now drive modern-day research, including solitons, chaos, stability, blow-up and singularity formation, asymptotic properties, etc., remained undetected or at best dimly perceived in the pre-computer era. The last sixty years has witnessed a remarkable blossoming in our understanding, due in large part to the insight offered by the availability of high performance computers coupled with great advances in the understanding and development of suitable numerical approximation schemes. New analytical methods, new mathematical theories, coupled with new computational algorithms have precipitated this revolution in our understanding and study of nonlinear systems, an activity that continues to grow in intensity and breadth. Each leap in computing power coupled with theoretical advances has led to yet deeper understanding of nonlinear phenomena, while simultaneously demonstrating how far we have yet to go. To make sense of this bewildering variety of methods, equations, and results, it is essential build upon a firm foundation on, first of all, linear systems theory, and secondly, nonlinear algebraic equations and nonlinear ordinary differential equations. Our presentation is arranged according to the order of the underlying differential equation. First order nonlinear partial differential equations model nonlinear waves and arise in gas dynamics, water waves, elastodynamics, chemical reactions, transport of pollutants, flood waves in rivers, chromatography, traffic flow, and a wide range of biological and ecological systems. One of the most important nonlinear phenomena, with no linear counterpart, is the break down of solutions in finite time, resulting in the formation of