  In this article we are concerned with an inverse initial boundary value
problem for a non-linear wave equation in space dimension $n\geq 2$. In
particular we consider the so called interior determination problem. This
non-linear wave equation has a trivial solution, i.e. zero solution. By
linearizing this equation at the trivial solution, we have the usual linear
wave equation with a time independent potential. For any small solution
$u=u(t,x)$ of this non-linear equation, it is the perturbation of linear wave
equation with time-independent potential perturbed by a divergence with respect
to $(t,x)$ of a vector whose components are quadratics with respect to
$\nabla_{t,x} u(t,x)$. By ignoring the terms with smallness $O(|\nabla_{t,x}
u(t,x)|^3)$, we will show that we can uniquely determine the potential and the
coefficients of these quadratics by many boundary measurements at the boundary
of the spacial domain over finite time interval and the final overdetermination
at $t=T$. In other words, the measurement is given by the so-called the
input-output map (see (1.5)).
