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Lecture Note Sketches Hermann Riecke - ESAM Home Page

Lecture Note Sketches Hermann Riecke - ESAM Home Page

∗ fronts Ψ+ and Ψ−

∗ fronts Ψ+ and Ψ− can interact and form a stable pair: wide localized wave train . A = 0 x L A = A c Figure 14: Localized wave trains in convection of water-alcohol mixtures. Space-time plot of the envelope of the left- (top panel) and the right-traveling wave component (bottom panel). In this parameter regime the localized convection waves are not stable but evolve chaotically (‘dispersive chaos’) [4, 5]. 86 x R

4 Fronts and Their Interaction Consider nonlinear PDEs with spatial translation symmetry that have multiple stable spatially homogeneous solutions ψ • ⇒ there must be also solutions that connect the stable states: fronts or kinks • these fronts are heteroclinic in space: they connect two different fixed points for x → ±∞. They are topologically stable: they cannot disappear except at infinity or by collision with ‘anti-fronts’. • This is to be compared to homoclinic solutions which connect to the same fixed point for x → ±∞, i.e. localized ‘humps’ (like the solitons). They are not topologically stable since they can disappear, e.g., due to a sufficiently large perturbation. stable stable unstable Figure 15: Fronts connecting two stable and one unstable spatially homogeneous state. Questions: • Do such fronts travel? What determines their speed? • How do the fronts interact? Can they from stable bound states: localized domains? 87 x

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