Nonlinear differential equations as invariants under group action on coset bundles: Burgers and Korteweg-de Vries equation families

Given a group, its coset spaces provide all homogeneous spaces for its action. A subgroup chain allows for the construction of a bundle of sections over a coset space of independent variables, where the fiber coordinates are dependent variables and all their partial derivatives up to some order, (i.e., the kth order jet). In this coset bundle, group invariants take the form of differential equations. We present two families of group-subgroup chains, one leading to various tensor Burgers-type differential equations, and the other to Korteweg-de Vries equations with an nth space derivative. Maps of the Hopf-Cole type appear in both families as transformations which intertwine the original group action to a multiplier realization of a normally extended group, yielding a new differential equation with greater symmetry.