Can we solve this ODE on $\mathbb{R}$ (it's the Fourier transform to the Burgers equation): $$\frac{d}{dt}\hat{u}+\frac{i y}{2}(\widehat{u^2})=-\epsilon y^2 \hat{u}$$ with the initial condition: $$\hat{u}(0,y)=-\frac{1}{y}$$ the $\hat{.}$ is the Fourier transform.
Is the non-linearity of the equation implies the it has no solution ?
As $\widehat{u^2}=C\hat u*\hat u$, the Fourier components do not really decouple, they stay connected via the convolution product. So what you have is still a differential equation over a function space, not over $\Bbb R$. So you still need discretization, now in the frequency domain, to get towards a numerical treatment.