I've been working on the following problem but I can't justify my steps, would a savvy mathematician kindly tell me what, if any, violations I've made.
Problem: Show Burgers' equation can be written as a system of infinitely many coupled ODE's. $$u_t+uu_x=\nu u_{xx}$$ and $$ \hat u_n=\frac{1}{2 \pi} \int_{\substack{-\pi\\}}^\pi u(x,t)e^{-ikx}dx$$ is the $n^{th}$ Fourier coefficient of $u(x,t)$. The boundaries are periodic.
First, I Fourier transform the equation to $$ \frac{d\hat u}{dt}-i\omega^2g(n) \hat u-\omega^2\nu \hat u =0$$ Next, I rewrite the Fourier transform of $u$ which I think takes care of the periodic boundary conditions by writing it as an infinite sum...this move feels sketchy: $$\hat u=\frac{1}{2\pi} \sum\limits_{n=-\infty}^\infty u(x,t)e^{-inx}$$ I then rewrite the Fourier transformed equation in terms of the infinite sums, where I cancel out the $\frac{e^{-inx}}{2\pi}$ terms and move the sum outside of everything to get $$ \sum\limits_{n=-\infty}^\infty (\frac{du}{dt}-u\omega^2(ig_n-\nu)=0)$$ which looks like a system of infinitely many coupled ODE's...but I would appreciate insight into whether it's mathematically correct or just plain wrong.