Math 639 — Nonlinear Waves — Syllabus

Linear waves, dissipation, dispersion. Conservation laws. Water waves. KdV equation, solitary waves, cnoidal waves. Scattering and inverse scattering. Perturbation theory. Nonlinear Schroedinger equation, dark and bright solitons, vortex solutions. Variational techniques, modulational instability, stability.

1. One of:
   1. Math 531 (Partial Differential Equations), or
   2. Math 537 (Ordinary Differential Equations).

The study of nonlinear systems has quietly and steadily revolutionized the realm of science over recent years. It is known that for nonlinear systems new structures emerge that have their features and peculiar ways of interacting. Examples of such structures abound in nature and include: vortices (like tornadoes or eddies in water tanks), solitons (bits of information used in optical fiber communications, water waves, tsunamis, humps of coherent matter waves, etc, ...), spirals (biological aggregates and chemical reactions). This course is intended as an introduction to the theory and of Nonlinear Waves and their applications. The course is intended for senior undergraduate and graduate students in Applied Mathematics, Computational Science, Engineering, Physics, Chemistry, Biology, etc. Examples from interdisciplinary areas will be covered. Most of the concepts and examples will be supplemented with hands-on computer codes.

• Conservation Laws: We will derive the core conservation laws of fluid mechanics from both an Eulerian and Lagrangian perspective. Bernoulli's equation will be derived and basic physical intuition in fluids will be developed through its use to solve fluid flow problems. Scaling will be introduced, and you will rescale the core conservation equations in several different regimes and explore the impact of non-dimensional parameters.
• Vorticity: The impact of vorticity on flows will be determined through the use of point-vortex approximations. Connections with differential equations will be derived as well as core theoretical results such as Kelvin and Helmholtz's Theorems. The definitions of baroclinic and barotropic fluids will also be introduced.
• Potential Theory: The use of analytic function theory to determine inviscid, incompressible, irrotational flows will be presented in several different geometries. The formulas for finding drag and lift on a body will be derived and used to compute profiles for a given body.
• Fourier Analysis for PDE's: We will solve linear PDE's using Fourier Series/Transforms. The concepts of group and phase velocity and dispersion will be explored and used to show how energy is distributed a dispersive PDE.
• Free Boundary Value Problems: Techniques for succinctly deriving evolution equations in the context of oceanic-free surface flows will be developed. This will make use of core results in vector calculus, such as the Divergence Theorem.
• Multiple Scales Analysis: The use of multiple scale ansatzes to reduce complex nonlinear models to simpler ones over smaller scales will be developed and used to solve several problems in fluids.
• Deriving nonlinear wave equations: The Korteweg de-Vries and Nonlinear Schroedinger equations will be derived. Solutions will be derived and their properties and implications for oceanic flows will be examined.

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• copying, in part or in whole, from another's test or other examination
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• replacing words or phrases from another source and inserting your own words or phrases

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