Propagation of Solitons Under Colored Noise
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1 Propagation of Solitons Under Colored Noise Dr. Russell Herman Departments of Mathematics & Statistics, Physics & Physical Oceanography UNC Wilmington, Wilmington, NC January 6, 2009
2 Outline of Talk 1 Solitary Waves and Solitons 2 White Noise and Colored Noise 3 Exact Solutions of the Stochastic KdV 4 Numerical Results to date 5 Summary Dr. Herman (UNCW) Stochastic Solitons JMM / 25
3 Solitary Waves and Solitons Solitary Waves The propagation of non-dispersive energy bundles through discrete and continuous media. Example - Burgers Equation u t + αuu x + βu xx = 0, u(x, t) = c α [1 + β tanh c 2 (x ct) ]. Dr. Herman (UNCW) Stochastic Solitons JMM / 25
4 Solitary Waves and Solitons Solitary Waves The propagation of non-dispersive energy bundles through discrete and continuous media. Example - Burgers Equation u t + αuu x + βu xx = 0, u(x, t) = c α [1 + β tanh c 2 (x ct) ]. Solitons Traveling wave solutions satisfying 1 They are of permanent form; 2 They are localised within a region; 3 They can interact with other solitons, and emerge from the collision unchanged, except for a phase shift. Example - KdV Equation u t + 6uu x + u xxx = 0, u(x, t) = 2η 2 sech 2 η(x 4η 2 t), c = 4η 2. Dr. Herman (UNCW) Stochastic Solitons JMM / 25
5 Evolution of One Soliton Solution Dr. Herman (UNCW) Stochastic Solitons JMM / 25
6 Evolution of One Soliton Solution Dr. Herman (UNCW) Stochastic Solitons JMM / 25
7 Evolution of One Soliton Solution Dr. Herman (UNCW) Stochastic Solitons JMM / 25
8 Evolution of One Soliton Solution Dr. Herman (UNCW) Stochastic Solitons JMM / 25
9 Evolution of One Soliton Solution Dr. Herman (UNCW) Stochastic Solitons JMM / 25
10 Evolution of One Soliton Solution Dr. Herman (UNCW) Stochastic Solitons JMM / 25
11 Evolution of One Soliton Solution Dr. Herman (UNCW) Stochastic Solitons JMM / 25
12 Evolution of One Soliton Solution Dr. Herman (UNCW) Stochastic Solitons JMM / 25
13 Evolution of One Soliton Solution Dr. Herman (UNCW) Stochastic Solitons JMM / 25
14 Evolution of One Soliton Solution Dr. Herman (UNCW) Stochastic Solitons JMM / 25
15 The Two Soliton Solution of the KdV Equation Form of the Solution When two solitons collide, they interact elastically. The exact solution for the two soliton equation is given by u(x, t) = 2(p2 q 2 )(p 2 + q 2 sech 2 χ(x, t)sinh 2 θ(x, t)) (p coshθ(x, t) q tanhχ(x, t)sinh θ(x, t)) 2 (1) where the phases are and θ(x, t) = px 4p 3 (t t 0 ) (2) χ(x, t) = qx 4q 3 (t t 0 ). (3) In our simulation we take p = 2, q = 1.5 and t 0 = 0.5. Dr. Herman (UNCW) Stochastic Solitons JMM / 25
16 Animation of Two Solitons Colliding Dr. Herman (UNCW) Stochastic Solitons JMM / 25
17 Animation of Two Solitons Colliding Dr. Herman (UNCW) Stochastic Solitons JMM / 25
18 Animation of Two Solitons Colliding Dr. Herman (UNCW) Stochastic Solitons JMM / 25
19 Animation of Two Solitons Colliding Dr. Herman (UNCW) Stochastic Solitons JMM / 25
20 Animation of Two Solitons Colliding Dr. Herman (UNCW) Stochastic Solitons JMM / 25
21 Animation of Two Solitons Colliding Dr. Herman (UNCW) Stochastic Solitons JMM / 25
22 Animation of Two Solitons Colliding Dr. Herman (UNCW) Stochastic Solitons JMM / 25
23 Animation of Two Solitons Colliding Dr. Herman (UNCW) Stochastic Solitons JMM / 25
24 Animation of Two Solitons Colliding Dr. Herman (UNCW) Stochastic Solitons JMM / 25
25 Animation of Two Solitons Colliding Dr. Herman (UNCW) Stochastic Solitons JMM / 25
26 The Two Soliton Solution of the KdV Equation Dr. Herman (UNCW) Stochastic Solitons JMM / 25
27 KdV-Burgers Equation Special Cases The KdV-Burgers Equation is given by u t + αuu x + βu xx + su xxx = 0. Traveling wave solutions are given for u(x, t) = 2k α [1 + tanh k(x 2kt)] (4) u(x, t) = 12sk2 α sech 2 k(x 4sk 2 t) (5) u(x, t) = Asech 2 η(x vt) + 2A[1 + tanhη(x vt)], (6) where A = 3β2 25αs, v = 6β2 5s, η = β 10s. Dr. Herman (UNCW) Stochastic Solitons JMM / 25
28 White Noise and Colored Noise Noise Types White - equal energy/cycle - constant frequency spectrum Pink - 1/f -noise - flat in log space - decreases 3 db per octave Brown - Decrease of 6 db per octave Blue - Increase 3 db per octave Purple - Increase 6 db per octave Dr. Herman (UNCW) Stochastic Solitons JMM / 25
29 White Noise and Colored Noise Noise Types White - equal energy/cycle - constant frequency spectrum Pink - 1/f -noise - flat in log space - decreases 3 db per octave Brown - Decrease of 6 db per octave Blue - Increase 3 db per octave Purple - Increase 6 db per octave Dr. Herman (UNCW) Stochastic Solitons JMM / 25
30 White Noise and Colored Noise Noise Types White - equal energy/cycle - constant frequency spectrum Pink - 1/f -noise - flat in log space - decreases 3 db per octave Brown - Decrease of 6 db per octave Blue - Increase 3 db per octave Purple - Increase 6 db per octave Dr. Herman (UNCW) Stochastic Solitons JMM / 25
31 White Noise and Colored Noise Noise Types White - equal energy/cycle - constant frequency spectrum Pink - 1/f -noise - flat in log space - decreases 3 db per octave Brown - Decrease of 6 db per octave Blue - Increase 3 db per octave Purple - Increase 6 db per octave Dr. Herman (UNCW) Stochastic Solitons JMM / 25
32 White Noise and Colored Noise Noise Types White - equal energy/cycle - constant frequency spectrum Pink - 1/f -noise - flat in log space - decreases 3 db per octave Brown - Decrease of 6 db per octave Blue - Increase 3 db per octave Purple - Increase 6 db per octave Dr. Herman (UNCW) Stochastic Solitons JMM / 25
33 Quantifying White Noise White Noise < η(t) >= 0. < η(t)η(s) >= δ(t s). η(t) is formal derivative of a Wiener process, W(t) Dr. Herman (UNCW) Stochastic Solitons JMM / 25
34 Quantifying White Noise White Noise < η(t) >= 0. < η(t)η(s) >= δ(t s). η(t) is formal derivative of a Wiener process, W(t) Brownian Motion dw dt = η(t) or dw = η(t)dt. Dr. Herman (UNCW) Stochastic Solitons JMM / 25
35 Quantifying White Noise White Noise < η(t) >= 0. < η(t)η(s) >= δ(t s). η(t) is formal derivative of a Wiener process, W(t) Brownian Motion dw dt = η(t) or dw = η(t)dt. Stochastic ODE dx dt = ǫη(t) dx = ǫdw. x i x i 1 = ǫ(w(t i ) W(t i 1 )) ǫdw i. Dr. Herman (UNCW) Stochastic Solitons JMM / 25
36 Quantifying Colored Noise Exponentially Correlated (Real) Noise - Ornstein-Uhlenbeck Process for dξ dt = 1 τ ξ + ǫ τ η(t) < ξ(t) >= 0, < ξ(t)ξ(s) >= ǫ2 2τ e t s /τ. Gaussian and stationary if < ξ(0) >= 0, < ξ 2 (0) >= ǫ2 2τ τ p(ξ 0 ) = ǫπ e ξ2 0 τ/ǫ Dr. Herman (UNCW) Stochastic Solitons JMM / 25
37 Quantifying Colored Noise Exponentially Correlated (Real) Noise - Ornstein-Uhlenbeck Process for dξ dt = 1 τ ξ + ǫ τ η(t) < ξ(t) >= 0, < ξ(t)ξ(s) >= ǫ2 2τ e t s /τ. Gaussian and stationary if < ξ(0) >= 0, < ξ 2 (0) >= ǫ2 2τ τ p(ξ 0 ) = ǫπ e ξ2 0 τ/ǫ 1/f β -noise, Kaulakys, et al., Physica A, 365 (2006) dx dt = Γx2σ 1 + x σ η(t), β = 2 2Γ + 1 2σ 2 Dr. Herman (UNCW) Stochastic Solitons JMM / 25
38 Exact Solution of the Stochastic KdV Equation Problem u t + 6uu x + u xxx = ζ(t), (7) Dr. Herman (UNCW) Stochastic Solitons JMM / 25
39 Exact Solution of the Stochastic KdV Equation Problem u t + 6uu x + u xxx = ζ(t), (7) Galilean transformation - Wadati 1983 u(x, t) = U(X, T) + W(T), X = x + m(t), T = t, (8) m(t) = 6 t W(t )dt, W(t) = t 0 0 transforms the stochastic KdV: U T + 6UU X + U XXX = 0. ζ(t )dt, Dr. Herman (UNCW) Stochastic Solitons JMM / 25
40 Exact Solution of the Stochastic KdV Equation Problem u t + 6uu x + u xxx = ζ(t), (7) Galilean transformation - Wadati 1983 u(x, t) = U(X, T) + W(T), X = x + m(t), T = t, (8) m(t) = 6 t 0 W(t )dt, W(t) = transforms the stochastic KdV: U T + 6UU X + U XXX = 0. The One Soliton Solution of the KdV t 0 ζ(t )dt, U(X, T) = 2η 2 sech 2 (η(x 4η 2 T X 0 )) (9) ( t )) u(x, t) = 2η 2 sech (η 2 x 4η 2 t x 0 6 W(t )dt + W(t). (10) 0 Dr. Herman (UNCW) Stochastic Solitons JMM / 25
41 Ensemble (Statistical) Average General < u(x, t) > = 4η2 π where a = 2η(x x 0 2η 2 t), = η 2 πb πk 2 sinhπk eiak bk dk. e (a s)2 /4b sech 2 s ds, (11) 2 b = 2η 2 < m 2 (t) >, t m(t) = 6 W(t )dt. 0 Dr. Herman (UNCW) Stochastic Solitons JMM / 25
42 Ensemble (Statistical) Average General < u(x, t) > = 4η2 π where a = 2η(x x 0 2η 2 t), = η 2 πb πk 2 sinhπk eiak bk dk. e (a s)2 /4b sech 2 s ds, (11) 2 b = 2η 2 < m 2 (t) >, t m(t) = 6 W(t )dt. 0 Results White Noise (Wadati 1983): b = 48η 2 ǫt 3 Exponential noise (Orlowski and Sobczyk 1989): ( 2 b = 36η 2 ǫ 2 τ 3 3 ( t τ )3 ( t τ )2 2(1 t ) τ )e t/τ + 2 Dr. Herman (UNCW) Stochastic Solitons JMM / 25
43 Averaged Soliton with Noise Dr. Herman (UNCW) Stochastic Solitons JMM / 25
44 Stochastic KdV: < u(x, t) > max vs. t Figure: 500 runs for x [ 10,90] with N = 1000, ǫ =.1, and η = 2. Dr. Herman (UNCW) Stochastic Solitons JMM / 25
45 Stochastic KdV: log(e 2γt/3 < u(x, t) > max ) vs. log t Figure: One Soliton Amplitude with Damping and Noise. Dr. Herman (UNCW) Stochastic Solitons JMM / 25
46 Stochastic KdV: < u(x, t) > max vs. t Figure: Two Soliton Solution with Damping and Noise. Dr. Herman (UNCW) Stochastic Solitons JMM / 25
47 What About Colored Noise vs White? 2.5 KdV Soliton Under White Noise 2 Amplitude <u(x,t)> Time t Figure: 500 runs for x [ 10, 90] with N = 500, ǫ =.1, and η = 2. Dr. Herman (UNCW) Stochastic Solitons JMM / 25
48 Stochastic KdV Amplitudes - Pink Noise 2.5 KdV Soliton Under Pink Noise 2 Amplitude <u(x,t)> Time t Figure: 1000 runs for x [ 10, 90] with N = 500, ǫ =.1, and η = 2. Dr. Herman (UNCW) Stochastic Solitons JMM / 25
49 Stochastic KdV Amplitudes - Pink Noise 2 KdV Soliton Under Pink Noise 1.5 Amplitude <u(x,t)> Time t Figure: 2000 runs for x [ 10, 90] with N = 500, ǫ =.1, and η = 2. Dr. Herman (UNCW) Stochastic Solitons JMM / 25
50 Stochastic KdV Amplitudes - Brown Noise KdV Soliton Under Brown Noise Amplitude <u(x,t)> Time t Figure: 1000 runs for x [ 10, 90] with N = 500, ǫ =.1, and η = 2. Dr. Herman (UNCW) Stochastic Solitons JMM / 25
51 Stochastic KdV Amplitudes - Brown Noise 2.5 KdV Soliton Under Brown Noise 2 Amplitude <u(x,t)> Time t Figure: 2000 runs for x [ 10, 90] with N = 500, ǫ =.1, and η = 2. Dr. Herman (UNCW) Stochastic Solitons JMM / 25
52 Stochastic KdV Amplitudes - Brown Noise 2.5 KdV Soliton Under Brown Noise 2 Amplitude <u(x,t)> Time t Figure: 3000 runs for x [ 10, 90] with N = 500, ǫ =.1, and η = 2. Dr. Herman (UNCW) Stochastic Solitons JMM / 25
53 Outline of Talk 1 Solitary Waves and Solitons 2 White Noise and Colored Noise 3 Exact Solutions of the Stochastic KdV 4 Numerical Results to date 5 Summary Dr. Herman (UNCW) Stochastic Solitons JMM / 25
54 Sample References Herman R L, 1990 The Stochastic, Damped KdV Equation J. Phys. A Herman R L and Rose A, 2008 Numerical Realizations of Solutions of the Stochastic KdV Equation, to appear in Mathematics and Computers in Simulation. Desmond J. Higham, An Algorithmic Introduction to Numerical Simulation of Stochastic Differential Equations, SIAM Review, vol. 43, no. 3, 2001, Kaulakys B et al., 2006, Nonlinear stochastic models of 1/f noise and power law distributions, Physicsa A 365, Miki Wadati, Stochastic Korteweg-de Vries Equation, Journal of the Physical Society of Japan, vol. 52, no. 8, August 1983, Miki Wadati and Yasuhiro Akutsu, Stochastic Korteweg-de Vries Equation with and without Damping, Journal of the Physical Society of Japan, vol. 53, no. 10, October 1984, Dr. Herman (UNCW) Stochastic Solitons JMM / 25
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