Nonlinearly dispersive nature of Galerkin-regularization and longon turbulence

With derivatives for physical insights and with mathematical analyses, technical variations and applications though, the dynamical nature of Galerkin truncation keeping finite Fourier modes in a nonlinear system is unclear. 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 other 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 linear dispersion functions with a large jump lead asymptotically to the GrBH dynamics.