PHYS 517   NONLINEAR  EVOLUTION  EQUATIONS AND SOLITONS

METU Credit (Theoretical-Laboratory hours/week):   3 (3-0)
ECTS Credit:     8.0
Department:     Physics
Course Coordinator:     AYŞE KARASU
Offered Semester:     Fall or Spring Semesters.


Course Objective:

Solitons are special solutions to equations which evolve in time with a fixed profile. Many equations have solitons, but multisolitons tend to occur in equations which are known as "integrable". Integrable equations are very rare and special because they can be solved analytically.They are heavily used in applied mathematics and mathematical physics. The aim of this elective course is to introduce the graduate students to integrable systems and solitons with emphasis on the main ideas that will be helpful for the beginners. By the end of the course unit, the students are expected to be able to apply some analytical methods for solving simple problems arising in nonlinear dynamical systems in one time and one space dimensions.


Course Contents:
  • Introduction-Wave phenomena, dispersion, dissipiation and nonlinearity.(2 weeks)
  • The Korteweg de Vries (KdV) equation-Derivation (Fermi Pasta Ulam Problem), properties,   elementary solutions.(3 weeks)
  • Further properties of KdV equation-Conservation laws, Lax formulation, Darboux transformations.(3 weeks)
  • Hamiltonian systems.(2 weeks)
  • Scattering and Inverse Scattering problems.(2 weeks)
  • The Painleve property.(2 weeks)

Textbook:

  " Solitons: an introduction ", by P.G.  Drazin and R.S. Johnson, Cambridge Univ. Press, 1989


References:
  • Linear and nonlinear waves, by G. B. Whitham, John Wiley, 1974.
  • Solitons and the inverse scattering transform, by M. Ablowitz and H. Segur, SIAM, 1981.
  • Chaos and integrability in nonlinear dynamics, by M. Tabor, John Wiley, 1989.
  • Wikipedia