spectral solution to 1D KdV equation - matlab

Question

I am trying to solve the solution of a 1D KdV equation (ut+uux+uxxx=0) starting from a two solitons initial condition using Fourier spectral method. My code blows up the solution for a reason that I cannot figure out. I would appreciate any help. This is my main code file:
%Spatial variable on x direction
L=4; %domain on x
delta=0.05; %spatial step size
xmin=-L; %minimum boundary
xmax=L; %maximum boundary
N=(xmax-xmin)/delta; %number of spatial points
x=linspace(xmin,xmax,N); %spatial vector
% 1D Initial state
sigma1 = 2; sigma2 = 4;
c1 = 2; c2 = 1;
h1 = 1; h2 = 0.3;
U = h1*sech(sigma2*(x+c1)).^2+h2*sech(sigma1*(x-c2)).^2;
%Fast Fourier Transform to the initial condition
Ut = fftshift(fft(U));
Ut = reshape(Ut,N,1);
%1D Wave vector disretisation
k = (2*pi/L)*[0:(N/2-1) (-N/2):-1];
k(1) = 10^(-6);
k = fftshift(k);
k = reshape(k,N,1);
%first derivative (advection)
duhat = 1i *k .*Ut;
du = real(ifft(ifftshift(duhat))); %inverse of FT
%third derivative (diffusion)
ddduhat = -1i * (k.^3) .*Ut;
dddu = real(ifft(ifftshift(ddduhat))); %inverse of FT
% Time variable
dt = 0.1; %time step
tspan = [0 4];
%solve
Time = 50;
for TimeIteration = 1:2:Time
t= TimeIteration * dt;
[Time,Sol] = ode45('FFT_rhs_1D',tspan,U,[],du,dddu);
Sol = Sol(TimeIteration,:);
%plotting
plot(x,abs(Sol),'b','LineWidth',2);
end
And this is the function that solves the equation:
function rhs = FFT_rhs_1D(tspan,U,dummy,du,dddu)
%solve the right hand side
rhs = - U .* du - dddu;
end
Thank you.