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Sergey A. Dyachenko
sergeydy AT buffalo DOT edu
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TR 12:30-1:50 pm at 225 Math
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T 3-4 Math Bldg 312 and by appointment
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G. B. Whitham, Linear and Nonlinear Waves.
Handwritten lecture notes supplemented the textbook throughout the course.
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This graduate selected-topics course develops mathematical ideas underlying
linear and nonlinear wave propagation. The course emphasizes the interaction
between dispersion and nonlinearity, asymptotic descriptions of dispersive waves,
and modulation methods. The presentation follows selected material from
Whitham's Linear and Nonlinear Waves, supplemented by handwritten
lecture notes and examples involving model wave equations.
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| 1--2 |
Introduction to wave propagation; linearization; dispersion relations; phase and group velocity. |
| 3--4 |
Fourier representation of linear dispersive waves; wave packets and dispersive evolution. |
| 5--6 |
Asymptotic evaluation of oscillatory integrals; method of stationary phase and large-time dispersive asymptotics. |
| 7--8 |
Nonlinear wave equations and the balance between nonlinearity and dispersion; representative model equations. |
| 9--10 |
Variational and Lagrangian formulations for wave equations; slowly varying wavetrains. |
| 11--12 |
Averaged Lagrangians and modulation of wavetrains; derivation of envelope equations. |
| 13 |
Schrödinger-type envelope equations and interpretation of dispersive coefficients. |
| 14 |
Nonlocal dispersive models, including the Benjamin--Ono equation and its dispersion relation. |
| 15 |
Selected topics in nonlinear waves and review/discussion. |
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The grade for the class is assigned at the end of the semester.
The written work is expected to be neat: illegible work will not be graded.
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