hwk10 - Phys 7268 Assignment 10 Due Problem 1 Exact...

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Phys. 7268 Assignment 10 Due: 4/14/11 Problem 1 Exact solution for a moving front. Compute analytically the shape and speed of the moving front connecting the saturated stripe solution with the uniform state for the (complex) type-I s amplitude equation. Hint: Use the fact that f ( x ) = tanh( x ) is a solution of the differential equation: f 00 + 2 f - 2 f 3 = 0 (1) Give the answer in dimensional (unscaled) form. Problem 2 Pulled Fronts in the Swift-Hohenberg Equation. (a) Show that for the Swift-Hohenberg equation t u ( x, t ) = ru - ( 2 x + 1) 2 u - u 3 (2) the velocity c of a pulled front and the stationary-phase wave number q s are given by c = 4 3 3 ( 2 + 1 + 6 r ) ( - 1 + 1 + 6 r ) 1 / 2 (3) and q s = 1 2 ( 3 + 1 + 6 r ) 1 / 2 + i 2 3 ( - 1 + 1 + 6 r ) 1 / 2 (4) (b) For small r verify that these results agree with the results expected from the amplitude equation approach. (c) Show that the wave number of the pattern laid down behind the advancing front, assuming no phase slips, is given by q fp = 3 ( 3 + 1 + 6 r ) 3 / 2 8 ( 2 + 1 + 6 r ) . (5) Problem 3 Qualitative analysis of fronts in a nonlinear diffusion equation.
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