On spatially non-local Burgers-like dynamical systems

The well-known hierarchy of the Burgers equation is equivalent to a hierarchy of non-local dynamical systems, which provide simple models for finite-distance spatial correlations or nearest-neighbour interactions in a physical situation. The new hierarchy is constructed from a sequence of Lax pairs. One member in each pair is the Forsyth–Hopf–Cole transformation. The second member is a linear equation, which is differential in time, with spatial delays in continuous space, with a discrete `spatial lag' λ. The dynamical equations (analogues of the Burgers equation hierarchy) are easily solved via the corresponding Lax pairs. For a given wave-propagation velocity, the solutions include single-, double- and triple-wavefronts. No higher-multiplicity wavefronts are generated. Finite-order approximations, obtained through the expansion of the dynamical equations of the new hierarchy in powers of λ, are compared with the explicit solutions. For a wide range of parameters, the low-order approximations are poor regardless of how small λ is, because of the singular nature of solutions of the dispersion relation.