A geometric approach to an equation of J. D’Alembert

Abstract:

By using a geometric framework of PDE’s we prove that the set of solutions of the D’Alembert equation $( \ast )(\frac {{{\partial ^2}\log f}}{{\partial x\partial y}}) = 0$ is larger than the set of smooth functions of two variables $f(x,y)$ of the form $( \ast \ast )f(x,y) = h(x) \bullet g(y)$. This agrees with a previous counterexample by Th. M. Rassias given to a statement by C. M. Stéphanos. More precisely, we have the following result. Theorem. The set of 2-dimensional integral manifolds of PDE $( \ast )$ properly contains the ones representable by graphs of 2-jet-derivatives of functions $f(x,y)$ expressed in the form $( \ast \ast )$. A generalization of this result to functions of more than two variables is sketched also by considering the equation $(\frac {{{\partial ^n}\log f}}{{\partial {x_1} \cdots \partial {x_n}}}) = 0$.