Nonlinear mathematical physics

• Know the properties of the fundamental nonlinear equations in physics
• Know the fundamental types of nonlinear solutions, such as bright and dark solitons
• Apply multiple-scale perturbation theory and Fredholm’s alternative for rigorous derivation and reduction of nonlinear equations
• Find conserved quantities and symmetries and use them to derive general nonlinear solutions
• Apply Virial theory to study collapse
• Calculate modulational instability gain bands for plane-wave solutions to nonlinear equations
• Describe soliton interaction
• Apply the Method of Moments to find solutions to nonlinear equations that do not have a Lagrangian
• Calculate when a soliton can emit linear dispersive waves and at what frequencies they appear
• Use Fourier-averaging perturbation theory to analyse second-harmonic generation and soliton properties in quadratic nonlinear materials with nonlinear quasi-phase-matching gratings and show how these gratings induce a cubic nonlinearity

Fundamental nonlinear equations (Sine-Gordon, Klein-Gordon, Korteweg-de-Vries, nonlinear Schrödinger, discrete nonlinear Schrödinger, and chi2 equations). Fundamental nonlinear solutions (bright, dark and solitons). Integrability. Multiple-scale perturbation theory. Method of moments. Virial theory. Fredholm’s alternative. Modulational instability. Soliton fission. Generation of dispersive waves – Cherenkov radiation. Raman redshift. Second harmonic generation, cascading, quasi-phase matching and induced Kerr nonlinearity.

34031

13 afternoons (1 – 5pm) starting with 3 lectures (35 minutes each) followed by group exercises (4 projects of 2 hours)

Minimum: 2.

Please be aware that this course will only be held if the required minimum number of participants is met. You will be informed 8 days before the start of the course, whether the course will be held.