Long-Timescale Soliton Dynamics in the Korteweg-de Vries Equation with Multiplicative Translation-Invariant Noise

Abstract

This paper studies the behavior of solitons in the Korteweg-de Vries equation under the influence of multiplicative noise. We introduce stochastic processes that track the amplitude and position of solitons based on a rescaled frame formulation and stability properties of the soliton family. We furthermore construct tractable approximations to the stochastic soliton amplitude and position which reveal their leading-order drift. We find that the statistical properties predicted by our method agree well with numerical evidence.

Figures (13)

• Figure 1: Simulation of the KdV equation with scalar noise of strength $\sigma=0.25$. Panel (a) shows the original frame realization $u(t,x)$, from a simulation of \ref{['eqn:scalar']}. Panel (b) shows $\phi_{c_*}(x)+v(t,x)$, from simulation in the frozen frame of \ref{['eqn:scalarmodulationv']}-\ref{['eqn:scalarmodulationxi']} with the same realization of the noise. Panels (c) and (d) show the perturbation with respect to the soliton, that is $u(t,x)-\phi_{c(t)}(x-\xi(t))$ with the phase-definitions \ref{['eqn:phasedefinitions']} in panel (c), and $v(t,x)$ in panel (d).
• Figure 2: Path-wise comparison of soliton amplitudes $c(t)$ to $c_{\text{fit}}(t)$ (left) and phase shifts $\Omega(t)$ to $\Omega_{\text{fit}}(t)$ (right) at noise strength $\sigma=0.25$ and initial amplitude $c_*=3$. The parameters $c_{\text{fit}}(t)$ and $\Omega_{\text{fit}}(t)$, defined in \ref{['eqn:phasedefinitions']} and \ref{['eqn:phaseshift']}, are obtained from direct simulation in the original frame of \ref{['eqn:scalar']}. The soliton amplitude $c(t)$ and phase shift $\Omega(t)$ are obtained from simulation of the frozen frame system \ref{['eqn:scalarmodulationv']}-\ref{['eqn:scalarmodulationxi']}.
• Figure 3: Sample mean of the process $\sup_{s\leq t}\|v(s)\|_{L_a^2([-50,20])}$ for scalar noise, see \ref{['eqn:scalarmodulationv']}, computed over $500$ realisations for $\sigma\in\{0.05,0.075,0.1,0.125\}$ and $c_*=3$. The exponential weight $e^{ax}$ in the $L^2_a$-norm strongly amplifies numerical effects entering from the right boundary of the computational domain $[-50,50]$. We take care to avoid these by computing the $L^2_a$-norm on $[-50,20]$, with $a=0.5$. For the initial soliton-parameter used in this simulation, the relevant dynamics occur well within $[-50,20]$ (see Figure \ref{['fig:pathwisesoliton']}).
• Figure 4: Sample mean of $v$ (dashed) and the approximation $v_2$ (solid) as the perturbation develops between $t=0.5$ and $t=2$. Computed over 3000 realisations for $\sigma=0.03$.
• Figure 5: Sample variance of the process $c_{\text{fit}}(t)/c_*$ for scalar noise and space-time white noise at various noise strengths $\sigma$. Solid lines indicate the sample variance, dashed lines indicate the variance of $c_0(t)$ as in \ref{['eqn:scalarvariance']} and \ref{['eqn:whitenoisevariance']}, respectively. The sample variance is computed over $204\cdot10^4$ and $8\cdot10^4$ realizations, respectively.

Theorems & Definitions (2)

• Lemma B.1
• proof