Spatio-Temporal Patterns in Nonlinear Systems

Syllabus

• Instabilities in physical systems: Rayleigh-Benard convection, self focusing and optical solitons, chemical oscillations, vegetation patterns.
• Mathematical analysis of instabilities: transcritical, saddle node, pitchfork and Hopf bifurcations.
• Primary instabilities in spatially extended systems: applications to convection and chemical oscillations.
• Amplitude equations: Non-Linear Shrodinger (NLS) equation, Newell-Whitehead-Segel (NWS) equation, Complex Ginzburg-Landau (CGL) equation.
• Secondary instabilities: Eckhaus, zigzag and Benjamin-Feir instabilities. Phase and amplitude turbulence.
• Front propagation in bistable systems: gradient vs. non-gradient systems, the non-equilibrium Ising-Bloch (NIB) bifurcation, transverse front instabilities.
• Singular perturbation theory of stationary and traveling patterns in bistable and excitable media.
• Spiral waves: kinematic theory, spiral turbulence.

Sources:

1. S.H. Strogatz, Nonlinear Dynamics and Chaos: with Applications to Physics, Biology, Chemistry and Engineering, (Addison-Wesley, Reading Mass., 1994). Q 172.5.C45S767.
2. P. Manneville, Dissipative Structures and Weak Turbulence, (perspectives in Physics, New York, 1990). QA 871 .M33.
3. A.C. Newell and J.V. Moloney, Nonlinear Optics, (Addison-Wesley, New York, 1992). QC446.2.N48.
4. M. Cross and P .C. Hohenberg, Pattern Formation outside of Equilibrium, Rev. Mod. Phys. 65 (1993) 2.
5. E. Meron, Pattern Formation in Excitable Media, Phys. Rep. 218 (1992) 1.