Nonlinearly dispersive nature of Galerkin-regularization and longon turbulence

With derivatives for physical insights and with mathematical analyses, technical variations and many applications though, the dynamical nature of Galerkin truncation in nonlinear systems is still not clear. Here, I show with such Galerkin-regularized Burgers-Hopf (GrBH) equation that the truncation corresponds to a nonlinear dispersion, supporting solitons and soliton-like structures (called "longons") and rhyming with recent expositions of dispersive objects. The formulation and scenarios resemble those of soliton turbulence, thus suggesting "longon turbulence" with large degree of freedoms (finite though). I also argue and numerically demonstrate that appropriate linearly dispersion models with an asymptotic large jump converge to the GrBH dynamics.