On the set of solutions of the generalized d'Alembert equation

By using a geometric approach we prove that the set of solutions of the generalized d'Alembert equation $\partial^n\log f/\partial x_1\cdots\partial x_n=0$, considered in the domain of the $(x^1,\cdots,x^n)$-space ${\mathbb R}^n$, is larger that the set of the functions that can be represented in the form as $ f(x^1,\cdots,x^n)=f_1(x^2,\cdots,x^n)\cdots f_n(x^1,\cdots,x^{n-1})$. Here the recent general method introduced by A. Pr\'astaro to calculate integral and quantum (co)bordism groups in PDE's is used. This method is very useful in order to prove existence of tunneling effects in PDE's, i.e., existence of solutions that change their sectional topology.