Title: Orbital Stability of Cnoidal Waves in the Korteweg-de Vries Equation against Localized Perturbations

Authors: Emile Bukieda, Björn de Rijk 

Sofware Version: MATLAB 24.1.0 R2024a, Mathematica 14.3.0.0

Background: The provided code was used to generate the plots in Figure 2 of the paper. It simulates the time evolution of a localized perturbation on top of a cnoidal wave.

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How to run: The code is run by executing the "Main.m" file. This then loads the initial perturbation stored in "Initial Perturbation.mat", sets up the simulation and computes the time evolution of the perturbation via "Strang_KdV.m". In the default, high resolution setting the runtime is around 2h.
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Initial perturbation: The construction of the initial perturbation is done in two steps. The neccesary files are located in the "Initial Perturbation" folder. First, in the Mathematica File "Two-Soliton on Cnoidal.nb", an approximate two-soliton solution on a cnoidal background is constructed and adjusted to the specs of the simulation. The output of this Mathematica file is the table-file "Two-Soliton on Cnoidal.csv". This solution, consisting of two well-separated approximate solitons on top of a cnoidal, is then evolved backwards in time until t=-4.5 in "Initial Perturbation.m", to produce a more localized initial perturbation. The final output is "Inital Perturbation.mat", which is the initial perturbation loaded by "Main.m".