Analysis of generalized quasilinear hyperbolic and Boussinesq equations from the point of view of potential symmetry (Q6636917)

Potential symmetries are determined for two partial differential equations in \((1+1)\)-dimensions, namely the so-called generalized quasilinear hyperbolic equations, given in terms of three arbitrary functions, and the Boussinesq equation. The method consists in rewriting the left-hand side of a given scalar equation \(\Delta[u]=0\), where \(\Delta[u]\) represents a differential function of \(u\) as a total divergence expression \(D_t A[u] + D_x B[u]\) for some differential functions \(A\) and \(B\) of \(u.\) A potential variable \(v=v(t,x)\) is then introduced by setting \(v_x= A[u]\) and \(v_t= -B[u],\) reducing the original equation into the latter system of two equations, referred to as the potential equations. A simple criterium for the existence of potential symmetries of the original equation among the symmetries of the corresponding potential equations is outlined. In the case of the family of hyperbolic equations, simplifying assumptions are made on the arbitrary parameter functions of the equation, leading to a number of potential symmetries in each case. But for the Boussinesq equation, several total divergence expressions are obtained for the same equation, none of which yields a potential symmetry. Although it is not mentioned whether the list of all the total divergence expressions derived for the equation is exhaustive, the authors conclude nevertheless that the Boussinesq equation does not admit potential symmetries.