Hierarchy of coupled Burgers-like equations induced by conditional symmetries

Abstract: It is known that $Q$-conditional symmetries of the classical Burgers' equation express in terms of three functions satisfying a coupled system of Burgers-like equations. The search of conditional symmetries of this system leads to a system of five coupled Burgers-like equations. Using the latter system as a starting point, and iterating the procedure, an infinite hierarchy of systems made of an odd number of coupled Burgers-like equations can be conjectured. Moreover, starting from a pair of Burgers-like equations, a similar hierarchy of systems made of an even number of coupled Burgers-like equations may arise. We prove that these two infinite hierarchies can be unified, and each element of the hierarchy arises from the nonclassical symmetries of the previous one. Writing a generic element of this hierarchy as a matrix Burgers' equation, the existence of the matrix Hopf-Cole transformation allows for its linearization and the determination of its solutions. Finally, it is shown that each element of the hierarchy possesses a five-dimensional Lie algebra of classical point symmetries. Though these Lie algebras are realized in manifolds with different dimensionality, they are all isomorphic.