Single-Course English 5 ECTS

Nonlinear mathematical physics

Overall Course Objectives

To introduce concepts, tools and methods from advanced nonlinear applied mathematics to physics and nanotechnology students and thereby enable them to study nonlinear effects in physics, optics, photonics, hydrodynamics, superconductivity, and biophysics.

Learning Objectives

  • 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

Course Content

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.

Recommended prerequisites

Teaching Method

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

Limited number of seats

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.

See course in the course database.





13 weeks




DTU Lyngby Campus

Course code 34451
Course type Candidate
Semester start Week 35
Semester end Week 48
Days Wed 13-17

7.500,00 DKK

Please note that this course has participants limitation. Read more