# Korteweg de Vries equation¶

Date: | 2018-10-24 (last modified), 2010-09-06 (created) |
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This page shows how the Korteweg-de Vries
equation can
be solved on a periodic domain using the method of
lines, with the
spatial derivatives computed using the pseudo-spectral method. In this
method, the derivatives are computed in the frequency domain by first
applying the FFT to the data, then multiplying by the appropriate values
and converting back to the spatial domain with the inverse FFT. This
method of differentiation is implemented by the **diff** function in the
module **scipy.fftpack**.

We discretize the spatial domain, and compute the spatial derivatives
using the **diff** function defined in the **scipy.fftpack** module. In
the following code, this function is given the alias **psdiff** to avoid
confusing it with the numpy function **diff**. By discretizing only the
spatial dimension, we obtain a system of ordinary differential
equations, which is implemented in the function **kdv(u, t, L)**. The
function **kdv_solution(u0, t, L)** uses **scipy.integrate.odeint** to
solve this system.

```
#!python
import numpy as np
from scipy.integrate import odeint
from scipy.fftpack import diff as psdiff
def kdv_exact(x, c):
"""Profile of the exact solution to the KdV for a single soliton on the real line."""
u = 0.5*c*np.cosh(0.5*np.sqrt(c)*x)**(-2)
return u
def kdv(u, t, L):
"""Differential equations for the KdV equation, discretized in x."""
# Compute the x derivatives using the pseudo-spectral method.
ux = psdiff(u, period=L)
uxxx = psdiff(u, period=L, order=3)
# Compute du/dt.
dudt = -6*u*ux - uxxx
return dudt
def kdv_solution(u0, t, L):
"""Use odeint to solve the KdV equation on a periodic domain.
`u0` is initial condition, `t` is the array of time values at which
the solution is to be computed, and `L` is the length of the periodic
domain."""
sol = odeint(kdv, u0, t, args=(L,), mxstep=5000)
return sol
if __name__ == "__main__":
# Set the size of the domain, and create the discretized grid.
L = 50.0
N = 64
dx = L / (N - 1.0)
x = np.linspace(0, (1-1.0/N)*L, N)
# Set the initial conditions.
# Not exact for two solitons on a periodic domain, but close enough...
u0 = kdv_exact(x-0.33*L, 0.75) + kdv_exact(x-0.65*L, 0.4)
# Set the time sample grid.
T = 200
t = np.linspace(0, T, 501)
print("Computing the solution.")
sol = kdv_solution(u0, t, L)
print("Plotting.")
import matplotlib.pyplot as plt
plt.figure(figsize=(6,5))
plt.imshow(sol[::-1, :], extent=[0,L,0,T])
plt.colorbar()
plt.xlabel('x')
plt.ylabel('t')
plt.axis('normal')
plt.title('Korteweg-de Vries on a Periodic Domain')
plt.show()
```

*Section author: WarrenWeckesser, Warren Weckesser*

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