Are Solitary Waves Color Blind to Noise?

Are Solitary Waves Color Blind to Noise? Dr. Russell Herman Department of Mathematics & Statistics, UNCW March 29, 2008

Outline of Talk 1 Solitary Waves and Solitons 2 White Noise and Colored Noise? 3 Quantifying Colored Noise 4 The Exact Solution of Stochastic KdV Equation 5 Numerical Results to date 6 Summary Dr. Herman (UNCW) Stochastic Solitons MAA-SE 2008 2 / 28

What Are Solitary Waves? Definition The propagation of non-dispersive energy bundles through discrete and continuous media. Dr. Herman (UNCW) Stochastic Solitons MAA-SE 2008 3 / 28

What Are Solitary Waves? Definition The propagation of non-dispersive energy bundles through discrete and continuous media. Example - Burgers Equation Let u(x, t) = f (x ct). Then, This yields a solution of the form u(x, t) = c α u t + αuu x + βu xx = 0 ( c + αf )f + βf = 0. [1 + β tanh c 2 (x ct) ]. Dr. Herman (UNCW) Stochastic Solitons MAA-SE 2008 3 / 28

Burgers Equation Traveling Wave Dr. Herman (UNCW) Stochastic Solitons MAA-SE 2008 4 / 28

What Are Solitons? History 1834 John Scott Russell s Great Wave of Translation 50 years of Controversy with Airy, Stokes, et al. 1870 s Boussinesq and Rayleigh verification 1895 Korteweg and devries derived PDE 1965 Zabusky & Kruskal revived KdV Equation in study of Fermi-Pasta-Ulam Problem Dr. Herman (UNCW) Stochastic Solitons MAA-SE 2008 5 / 28

What Are Solitons? History Soliton 1834 John Scott Russell s Great Wave of Translation 50 years of Controversy with Airy, Stokes, et al. 1870 s Boussinesq and Rayleigh verification 1895 Korteweg and devries derived PDE 1965 Zabusky & Kruskal revived KdV Equation in study of Fermi-Pasta-Ulam Problem 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. Dr. Herman (UNCW) Stochastic Solitons MAA-SE 2008 5 / 28

The KdV Equation The PDE The Korteweg-deVries Equation takes several forms. u t + αuu x + βu xxx = 0. For example, u t + 6uu x + u xxx = 0. The nonlinear term can balance the dispersive term. One - Soliton Solution Let u(x, t) = f (x ct). Then, ( c + 6f )f + f = 0. This yields a solution of the form u(x, t) = 2η 2 sech 2 η(x 4η 2 t), c = 4η 2. Dr. Herman (UNCW) Stochastic Solitons MAA-SE 2008 6 / 28

Evolution of One Soliton Solution Dr. Herman (UNCW) Stochastic Solitons MAA-SE 2008 7 / 28

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 cosh θ(x, t) q tanh χ(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 MAA-SE 2008 8 / 28

Animation of Two Solitons Colliding Dr. Herman (UNCW) Stochastic Solitons MAA-SE 2008 9 / 28

The Two Soliton Solution of the KdV Equation Dr. Herman (UNCW) Stochastic Solitons MAA-SE 2008 10 / 28

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) = A sech 2 η(x vt) + 2A [1 + tanh η(x vt)], (6) where A = 3β2 25αs, v = 6β2 5s, η = β 10s. Dr. Herman (UNCW) Stochastic Solitons MAA-SE 2008 11 / 28

KdV-Burgers Traveling Wave Solution Dr. Herman (UNCW) Stochastic Solitons MAA-SE 2008 12 / 28

What is 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 MAA-SE 2008 13 / 28

What is 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 MAA-SE 2008 14 / 28

What is 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 MAA-SE 2008 14 / 28

What is 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 = ɛη(t) dx = ɛdw. x i x i 1 = ɛ(w (t i ) W (t i 1 )) ɛdw i. Dr. Herman (UNCW) Stochastic Solitons MAA-SE 2008 14 / 28

Quantifying Colored Noise Real Systems Are Not White where dx dt = αx + γ + µxξ(t), < ξ(t)ξ(s) >= Ae t s /τ. Dr. Herman (UNCW) Stochastic Solitons MAA-SE 2008 15 / 28

Quantifying Colored Noise Real Systems Are Not White where Ornstein-Uhlenbeck Process for dx dt = αx + γ + µxξ(t), < ξ(t)ξ(s) >= Ae t s /τ. dξ dt = 1 τ ξ + ɛ τ η(t) < ξ(t) >= 0, < ξ(t)ξ(s) >= ɛ2 2τ e t s /τ. Dr. Herman (UNCW) Stochastic Solitons MAA-SE 2008 15 / 28

Exact Solution of Stochastic KdV - Wadati - 1983 Time-dependent SkDV u t + 6uu x + u xxx = ζ(t), (7) ζ(t) is Gaussian white noise: zero mean and (< >= E[ ]) < ζ(t)ζ(t ) >= 2ɛδ(t t ). (8) Dr. Herman (UNCW) Stochastic Solitons MAA-SE 2008 16 / 28

Exact Solution of Stochastic KdV - Wadati - 1983 Time-dependent SkDV u t + 6uu x + u xxx = ζ(t), (7) ζ(t) is Gaussian white noise: zero mean and (< >= E[ ]) < ζ(t)ζ(t ) >= 2ɛδ(t t ). (8) Galilean transformation u(x, t) = U(X, T ) + W (T ), X = x + m(t), T = t, (9) m(t) = 6 t W (t )dt, W (t) = t 0 0 ζ(t ) dt, transforms the stochastic KdV: U T + 6UU X + U XXX = 0, (10) Dr. Herman (UNCW) Stochastic Solitons MAA-SE 2008 16 / 28

The One Soliton Solution Under Noise The One Soliton Solution of the KdV U(X, T ) = 2η 2 sech 2 (η(x 4η 2 T X 0 )) (11) Leads directly to exact skdv solution: ( t )) u(x, t) = 2η 2 sech (η 2 x 4η 2 t x 0 6 W (t ) dt + W (t). (12) 0 http://people.uncw.edu/hermanr/research/skdv/soliton_movie.htm Dr. Herman (UNCW) Stochastic Solitons MAA-SE 2008 17 / 28

The One Soliton Solution Under Noise The One Soliton Solution of the KdV U(X, T ) = 2η 2 sech 2 (η(x 4η 2 T X 0 )) (11) Leads directly to exact skdv solution: ( t )) u(x, t) = 2η 2 sech (η 2 x 4η 2 t x 0 6 W (t ) dt + W (t). (12) 0 http://people.uncw.edu/hermanr/research/skdv/soliton_movie.htm Exact Statistical Average where < u(x, t) > = 4η2 π = η 2 πb a = 2η(x x 0 2η 2 t), πk 2 sinh πk eiak bk dk. e (a s)2 /4b sech 2 s ds, (13) 2 Dr. Herman (UNCW) Stochastic Solitons MAA-SE 2008 17 / 28

Averaged Soliton with Noise Dr. Herman (UNCW) Stochastic Solitons MAA-SE 2008 18 / 28

Stochastic KdV Amplitudes - < u(x, t) > max vs. t - Exact Comparison Figure: 500 runs for x [ 10, 90] with N = 1000, ɛ =.1, and η = 2. Dr. Herman (UNCW) Stochastic Solitons MAA-SE 2008 19 / 28

Stochastic KdV Amplitudes - White Figure: 500 runs for x [ 10, 90] with N = 500, ɛ =.1, and η = 2. Dr. Herman (UNCW) Stochastic Solitons MAA-SE 2008 20 / 28

Stochastic KdV Amplitudes - Pink Noise Figure: 1000 runs for x [ 10, 90] with N = 500, ɛ =.1, and η = 2. Dr. Herman (UNCW) Stochastic Solitons MAA-SE 2008 21 / 28

Stochastic KdV Amplitudes - Pink Noise Figure: 2000 runs for x [ 10, 90] with N = 500, ɛ =.1, and η = 2. Dr. Herman (UNCW) Stochastic Solitons MAA-SE 2008 22 / 28

Stochastic KdV Amplitudes - Brown Noise Figure: 1000 runs for x [ 10, 90] with N = 500, ɛ =.1, and η = 2. Dr. Herman (UNCW) Stochastic Solitons MAA-SE 2008 23 / 28

Stochastic KdV Amplitudes - Brown Noise Figure: 2000 runs for x [ 10, 90] with N = 500, ɛ =.1, and η = 2. Dr. Herman (UNCW) Stochastic Solitons MAA-SE 2008 24 / 28

Stochastic KdV Amplitudes - Brown Noise Figure: 3000 runs for x [ 10, 90] with N = 500, ɛ =.1, and η = 2. Dr. Herman (UNCW) Stochastic Solitons MAA-SE 2008 25 / 28

So, What is the answer? What s the question? Are Solitary Waves Color Blind to Noise? Dr. Herman (UNCW) Stochastic Solitons MAA-SE 2008 27 / 28

So, What is the answer? What s the question? Are Solitary Waves Color Blind to Noise? Answer Most likely - NO Thank you. Dr. Herman (UNCW) Stochastic Solitons MAA-SE 2008 27 / 28

Sample References R. L. Herman, Solitary Waves, American Scientist, vol. 80, July-August 1992, 350-361. R. L. Herman and C. J. Knickerbocker, Numerically Induced Phase Shift in the KdV Soliton, Journal of Computational Physics, vol. 104, no. 1, January 1993, 50-55. Desmond J. Higham, An Algorithmic Introduction to Numerical Simulation of Stochastic Differential Equations, SIAM Review, vol. 43, no. 3, 2001, 525-546. A. C. Vliegenthart, On Finite-Difference Methods for the Korteweg-de Vries Equation, Journal of Engineering Mathematics, vol. 5, no. 2, April 1971, 137-155. Miki Wadati, Stochastic Korteweg-de Vries Equation, Journal of the Physical Society of Japan, vol. 52, no. 8, August 1983, 2642-2648. 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, 3342-3350. Dr. Herman (UNCW) Stochastic Solitons MAA-SE 2008 28 / 28