Short Course Description
Course "Asymptotic methods"
Instructor: Boris Melamed (Malomed)
Teaching language: English
The grade will be given solely on the basis of a final exam (solving a set of six problems on basic parts of the course)
Brief syllabus:
Topic 1: Asymptotic methods for real and complex algebraic equations with small or large parameters. Regular and singular perturbations. Asymptotic series vs. the regular Taylor expansion.
Topic 2: Approximate calculation of integrals. Integrals of rapidly oscillating functions. Stationary-phase points (applications to Airy functions etc.). The steepest descent approximation.
Topic 3: Asymptotic methods for ordinary differential equations. The averaging method for rapidly oscillating solutions. The Hopf bifurcation, limit cycles, and the phase-plane methods. Nonlinear oscillators and the theory of directly driven and parametric resonances in linear, nonlinear, and dissipative systems. Dynamical chaos and strange attractors. The concept of Hamiltonian chaos and the Kolmogorov-Arnold-Moser theorem.
Topic 4: Asymptotic methods for partial differential equations. Complex Ginzburg-Landau and nonlinear Schroedinger equations. Plane waves and modulational instability. Fronts, shock waves, and the geometric-optics (eikonal) approximation. The concept of solitons (solitary waves) and the perturbation theory for solitons. Basic physical realizations of nonlinear partial differential equations.
Remark: some parts of the course may be curtailed if the duration of the semester is insufficient.
Recommended books
Main: A. Nayfeh, Perturbation Methods (Wiley, 2004).
Additional: J. C. Neu, Singular Perturbation in the Physical Sciences )American Mathematical Society, 2015);
Y. S. Kivshar and B. A. Malomed, Dynamics of solitons in nearly integrable systems. Rev. Mod. Phys. 61, 763-915 (1989).
Preliminary requirements: introductory courses of classical mechanics, ordinary differential equations, and partial differential equations.
Full Syllabus
