The course presents several powerful techniques for solving mathematical physics problems. The techniques are based on the use of expansions on a small parameter present in the equation under consideration or introduced by a suitable rescaling. The development of these methods leads to important mathematical concepts such as the Maslov index and the Lagrange variety. The methods are illustrated by solving various classical problems from quantum mechanics and wave propagation theory. The course includes a practical part where students will develop simple numerical models to solve wave propagation problems in realistic scenarios.

Prerequisites: Linear Algebra, Analysis in Rn, Ordinary Differential Equations

1. differential equations with a small parameter.
2. classical WKB method, applications in quantum mechanics and acoustics.
3. semiclassical Einstein-Brillouin-Keller quantization methods.
4. WKB approximation with rays, ray congruences.
5. Gaussian beams.
6. multiscale method.
7. parabolic equations for wave modeling.
8. Canonical operator, Maslov index, Lagrange varieties.
9. Calculation of canonical operators for specific problems.

References:
1. Nayfeh A.H. Perturbation methods. John Wiley & Sons, 2024.
2. Popov M.M. Ray theory and Gaussian beam method for geophysicists. Editora da Universidade Federal da Bahia, 2002.
3. Trofimov M., Kozitskiy S., Zakharenko A., Petrov P. Formal derivations of mode coupling equations in underwater acoustics: how the method of multiple scales results in an expansion over eigenfunctions and the vectorized wkbj solution for the amplitudes// Journal of Marine Science and Engineering, 2023, Vol. 11(4), Art. No. 797.
4. Petrov P. S., Ehrhardt M., Makarov, D. V. Multiscale Approach to Parabolic Equations Derivation: Beyond the Linear Theory// Procedia Computer Science, 2017, Vol. 108, Pp. 1823-1831.

Note: This course is offered as a master’s course. In the doctorate, it has additional requirements.

 

* Basic syllabus. The teacher has the autonomy to make any changes.