Solving a system of ODE that arose in solving Burgers' equation Consider the Burgers' equation $$\partial_t u = \alpha u\partial_xu$$ Intend to solve this using Fourier Galerkin method. So When I convert this into $N$th Fourier partial sum, I get a system of ODE's as follows $$\partial_t u_m = \sum_{k=-N}^{k=N}iku_ku_{m-k}    $$
How do I solve this system of ODE numerically or analytically?, given we know the first $N$ Fourier coefficients of the initial condition $u(x,0)$.

Analytic solving of the PDE :
$$\partial_t u - \alpha u\partial_xu=0$$
Thanks to the method of characteristics :
$$\frac{dt}{1}=\frac{dx}{-\alpha u}=\frac{du}{0}$$
On the characteristic curves : 
  \begin{cases}
    u=c_1\\
    \alpha c_1 dt +dx=0\:\:\rightarrow\:\: \alpha u t +x=c_2\\
  \end{cases}
The general solution can be expressed on the form : 
$$\Phi\left(u\:,\: \alpha t u +x\right)=0$$
where $\Phi$ is any derivable function of two variables. This is equivalent of the implicit relationship :
$$u=F\left(\alpha \;t\; u +x\right)$$
where $F$ is any derivable function. 
Nothing can be done further until the boundary conditions be defined, in order to find the convenient function $F$ and then $u(x,t)$ when it is possible to explicit $u$ from the implicit equation.
