  Spectral methods provide highly accurate numerical solutions for partial
differential equations, exhibiting exponential convergence with the number of
spectral nodes. Traditionally, in addressing time-dependent nonlinear problems,
attention has been on low-order finite difference schemes for time
discretization and spectral element schemes for spatial variables. However, our
recent developments have resulted in the application of spectral methods to
both space and time variables, preserving spectral convergence in both domains.
Leveraging Tensor Train techniques, our approach tackles the curse of
dimensionality inherent in space-time methods. Here, we extend this methodology
to the nonlinear time-dependent convection-diffusion equation. Our
discretization scheme exhibits a low-rank structure, facilitating translation
to tensor-train (TT) format. Nevertheless, controlling the TT-rank across
Newton's iterations, needed to deal with the nonlinearity, poses a challenge,
leading us to devise the "Step Truncation TT-Newton" method. We demonstrate the
exponential convergence of our methods through various benchmark examples.
Importantly, our scheme offers significantly reduced memory requirement
compared to the full-grid scheme.
