Solitons in the Korteweg-de Vries Equation

TL;DR

This work investigates numerical schemes for the Korteweg-de Vries (KdV) equation to simulate soliton dynamics, comparing an implicit Crank-Nicolson discretization with a Fast Fourier Transform (FFT) based approach. It analyzes single- and two-soliton scenarios, using predictor-corrector steps and periodic boundaries, and assesses accuracy via global $L^2$ error against analytical solutions. The study finds that Crank-Nicolson yields higher accuracy yet incurs substantially higher computational cost than FFT, and it validates soliton behavior and interactions (including merge-split and bounce-exchange) under the two schemes. The results highlight practical trade-offs between accuracy and speed and suggest extensions to boundary conditions, generalized KdV equations, and higher-order soliton configurations with alternative numerical methods.

Abstract

We propose a numerical solution to the Korteweg-de Vries (KdV) equation using a Crank-Nicolson scheme, and compare its performance to the Fast Fourier Transform method. The properties and interactions of soliton solutions are further examined. Initial conditions were varied to analyse soliton formation in the resulting system. Performing an L$^2$ error analysis demonstrated consistency between numerical methods of solving the KdV equation and analytical solutions.

Figures (6)

• Figure 1: In each illustrated case, the initial condition was taken to be an exact secant-squared soliton solution of the KdV equation with $t = 0$. Excluding the non-linear advection term, dispersion is observed as displayed on the left. In the graph on the right, eliminating the dispersion term illustrates the breaking effect of the advection term. For both graphs, wave height is plotted against position with dimensional units [L] for distance and [T] for time.
• Figure 2: Illustration of periodic boundary conditions applied to the discretised KdV equation for a single soliton evolving to the right from an initial condition of Equation \ref{['eq:5']} with $t = 0$ on a plot of position against amplitude. Note how the wave wraps around the boundary, appearing to continously re-emerge from the $x = 0$ boundary as it meets that at $x = L$ for the numerical solution as it does for the analytical.
• Figure 3: Down the left side, a Gaussian initial condition is seen evolving a primary wave and secondary wavelets travelling in the opposite direction for times t = 0.000, 0.740 and 1.480 [T] on a plot of amplitude against position. Down the right side, peak height, width and velocity tend towards constant average values following the decay of transient behaviour. The oscillating pattern results from interactions between the wavelets and the soliton as they pass the boundary with periodic conditions and intersect. Note $\Delta t = 10^{-3}$ [T] and $\Delta x = 0.05$ [L].
• Figure 4: Shown above in a plot of position versus amplitude is a merge-split interaction, corresponding to $r > 3$, between two solitons evolving to the right from a secant-squared initial condition (that is, Equation \ref{['eq:5']} with $t=0$). Step sizes of $\Delta x = 0.025$ and $\Delta t = 10^{-3}$ were used with a total error of 0.310 in the numerical solution compared to the analytical solution. A graph of local L$^2$ error recorded as a function of time is displayed. Note the dip which occurs during the interaction. The analytical solution was modelled by Equation \ref{['eq:6']}.
• Figure 5: Above is a bounce-exchange 2-soliton interaction, indicating $r < 3$, on a plot of amplitude against position. Each soliton was produced with a secant-squared initial condition (from Equation \ref{['eq:5']} with $t=0$) with a velocity in the positive x-direction, then stepped by $\Delta x = 0.025$ and $\Delta t = 10^{-3}$ to arrive at a global error of 0.336 [L$^\frac{3}{2}$]. The analytical solution was plotted using Equation \ref{['eq:6']}.