Analytical methods for solving nonlinear problems in science and engineering are discussed. A special attention is paid to the investigation of phenomena governed by ordinary and partial differential equations by means of the multi-scale approach. Examples from physics, chemistry, materials science and biology are considered.

Main topics:

• Multiple-scale approach for studying bifurcations and nonlinear oscillations. Van-der-Pol oscillator. Lindstedt-Poincare method.
• Nonlinear waves in fluid dynamics and optics. Nonlinear Schrödinger equation, its symmetries and exact solutions. Modulational instability. Korteweg-de Vries equation. Solitons.
• Instabilities in extended systems. Examples and classification. Reaction-diffusion systems. Patterns, fronts and domain walls. Pattern selection.
• Modulational instabilities of patterns. Ginzburg-Landau equation. Kuramoto-Sivashinsky equation. Spatio-temporal chaos.

Recommended books:

1. J. Kevorkian, J.D. Cole, Multiple scale and singular perturbation methods, Springer, New York, 1996.
2. R.K. Dodd et al., Solitons and nonlinear wave equations, Academic Press, London, 1984.
3. D. Walgraef, Spatio-temporal pattern formation: with examples from physics, chemistry, and material science, Springer, New York, 1997.
4. J.D. Murray, Mathematical biology, Springer, New York, 2001.

For more detailed information call (467-3451), stop by (M464), or send e-mail: alexn@jupiter.esam.northwestern.edu