Physical meaning of third derivative with respect to position I currently on a numerical solver for the KdV equation which reads 
$$ u_t + uu_x = u_{xxx} $$
I was wondering the physical sense of this third derivative with respect to $x$. I know that the $uu_x$ term is a convection term and that a second and fourth derivative modelize diffusion but I have no idea about the third derivative. 
I used my Fourier spectral code to solve it in order to understand and I obtained the following profile after one second of simulation :

Looks like the third derivative has the same effect has the convective term.
 A: The Korteweg-de Vries equation is a nonlinear dispersive wave equation which describes the amplitude of shallow water waves. The version of the equation you're quoting is the nondimensionalized and rescaled version. According to this article from Wolfram MathWorld the original equation reads
$$ \frac{\partial\eta}{\partial{t}}=\frac{3}{2}\sqrt\frac{g}{h}\left(\eta\frac{\partial\eta}{\partial{x}}+\frac{2}{3}\frac{\partial\eta}{\partial{}x}+\frac{1}{3}\sigma\frac{\partial^3\eta}{\partial{}x^3}\right)$$
As specified in the comments, the meaning of the third derivative is specific to the problem. Have a look at this derivation of the equation by Professor Axel Brandenburg from the University of Colorado if you want to understand the details of the equation.
To put it simply the third derivative is responsible for the dispersivity of the equation. If you add a scalar prefactor to this derivative you will observe a dispersion of your initial cosine wave. I found the following animation on the Wikepedia article of the equation. It's the numerical solution of $u_t+uu_x+\delta^2u_{xxx}=$ with $\delta=0.022$ and an initial condition $u(x,0)=\cos\pi{}x$.

If you want to run the simulation by yourself you can use the Python code below. This code is an adaptation of the MATLAB code from Llyod N. Trefehen (p27.m). It uses Fourier spectral method with an integrating prefactor and the Runge-Kutta scheme. A simple Fourier spectral code with the Euler implicit method will do the work too.
import numpy as np
import matplotlib.pyplot as plt
import matplotlib.animation as animation

def animate(i,uu,line):
    line.set_data(x,uu[100*i,:])
    return line,

a  = .022 # dispersion prefactor
N  = 256
tf = 2.
dt = .001
Nt = int(round(tf/dt))
x0 = -np.pi
xf =  np.pi
x  = (xf-x0)/N*np.arange(-N/2,N/2)
u  = np.cos(x) # initial condition
uu = np.zeros((Nt,N)); uu[0,:] = u

M = int(round(N/2))
k = np.zeros(N)
k[0:M] = np.arange(0,M); k[M+1:] = np.arange(-M+1,0,1)
ik3 = 1j*k**3
U = np.fft.fft(u) # go to spectral space
for n in range(1,Nt):
    t = n*dt
    g = -0.5j*dt*k
    expaik = np.exp(a*dt*ik3/2) # integrating
    expaik2 = expaik**2               # factor
    # Runge-Kutta scheme
    rk1 = g * np.fft.fft(np.real(np.fft.ifft(U))**2)
    rk2 = g * np.fft.fft(np.real(np.fft.ifft(expaik*(U+rk1/2)))**2)
    rk3 = g * np.fft.fft(np.real(np.fft.ifft(expaik*U+rk2/2))**2)
    rk4 = g * np.fft.fft(np.real(np.fft.ifft(expaik2*U+expaik*rk3))**2)
    U = expaik2*U + (expaik2*rk1 + 2*expaik*(rk2+rk3) + rk4)/6
    u = np.real(np.fft.ifft(U)) # back to real space
    uu[n,:] = u

fig = plt.figure()
ax  = fig.gca()
line, = ax.plot([],[])
ax.plot(x,uu[0,:])
ax.legend(['current time','initial'])
ax.set_xlim(x0,xf)
ax.set_ylim(-1,3)
ax.set_xlabel(r'$x$')
ax.set_ylabel(r'$u(x)$')
ax.grid(True)

anim = animation.FuncAnimation(fig,animate,frames=int(Nt/100),fargs=(uu,line),repeat=False)
anim.save('anim.mp4')
plt.show()


