Numerical realizations of solutions of the stochastic KdV equation

We investigate simulations of exact solutions of the stochastic Dr. Herman Numerical Realizations of Solutions f the Stochastic KdV Equation Exact Solution of Stochastic KdV Wadati 1983 The One Soliton Solution Under Noise Statistical Averages The Exact Solution for < u(x, t) > via the Diffusion Equation Solving the Diffusion Equation The Damped Stochastic KdV Asymptotics Numerical Simulation of the Stochastic KdV Simulating Brownian Motion Brownian Motion Simulation, Mean, and Variance Wadati Identity Numerical Results Code for Averaged Soliton Single Soliton Plus Noise Averaged Soliton with Noise Average Soliton Amplitude Loglog Plot for Average Soliton Amplitude Zabusky-Kruskal Scheme Plus Stochastic Terms Vectorized Form Stochastic KdV Results Damped, Stochastic KdV Amplitudes Varying N Comparisons of Amplitude Decay Due to Noise and Damping Amplitude Decay for Several Parameters Integrated Decay Constants for t ∈ [0, 10 Decay Constants for t ∈ [0, 100 Decay Constants for Integral Computation Decay Constants for Finite Difference Results The Two Soliton Solution of the KdV Equation Amplitude and Position of Damped Solitons Under Noise Amplitude and Position of Damped Solitons Under Noise Conclusions ut + 6uux + uxxx = ζ(t), (1) ζ(t) is Gaussian white noise: zero mean and (< ∗ >= E [∗]) Dr. Herman Numerical Realizations of Solutions of the Stochastic KdV Equation Exact Solution of Stochastic KdV Wadati 1983 The One Soliton Solution Under Noise Statistical Averages The Exact Solution for < u(x, t) > via the Diffusion Equation Solving the Diffusion Equation The Damped Stochastic KdV Asymptotics Numerical Simulation of the Stochastic KdV Simulating Brownian Motion Brownian Motion Simulation, Mean, and Variance Wadati Identity Numerical Results Code for Averaged Soliton Single Soliton Plus Noise Averaged Soliton with Noise Average Soliton Amplitude Loglog Plot for Average Soliton Amplitude Zabusky-Kruskal Scheme Plus Stochastic Terms Vectorized Form Stochastic KdV Results Damped, Stochastic KdV Amplitudes Varying N Comparisons of Amplitude Decay Due to Noise and Damping Amplitude Decay for Several Parameters Integrated Decay Constants for t ∈ [0, 10 Decay Constants for t ∈ [0, 100 Decay Constants for Integral Computation Decay Constants for Finite Difference Results The Two Soliton Solution of the KdV Equation Amplitude and Position of Damped Solitons Under Noise Amplitude and Position of Damped Solitons Under Noise Conclusions We consider the one soliton solution of the KdV: U(X ,T ) = 2η22(η(X − 4η2T − X0)) (6) Dr. Herman Numerical Realizations of Solutions of the Stochastic KdV Equation Exact Solution of Stochastic KdV Wadati 1983 The One Soliton Solution Under Noise Statistical Averages The Exact Solution for < u(x, t) > via the Diffusion Equation Solving the Diffusion Equation The Damped Stochastic KdV Asymptotics Numerical Simulation of the Stochastic KdV Simulating Brownian Motion Brownian Motion Simulation, Mean, and Variance Wadati Identity Numerical Results Code for Averaged Soliton Single Soliton Plus Noise Averaged Soliton with Noise Average Soliton Amplitude Loglog Plot for Average Soliton Amplitude Zabusky-Kruskal Scheme Plus Stochastic Terms Vectorized Form Stochastic KdV Results Damped, Stochastic KdV Amplitudes Varying N Comparisons of Amplitude Decay Due to Noise and Damping Amplitude Decay for Several Parameters Integrated Decay Constants for t ∈ [0, 10 Decay Constants for t ∈ [0, 100 Decay Constants for Integral Computation Decay Constants for Finite Difference Results The Two Soliton Solution of the KdV Equation Amplitude and Position of Damped Solitons Under Noise Amplitude and Position of Damped Solitons Under Noise Conclusions