Existence and Uniqueness for Periodic Solutions of the Benjamin–Bona–Mahony Equation

We consider the problem $u_t - u_{xx} + uu_x = 0$ in $ - \infty < x$, $t < \infty $ with initial data at $t = 0$ which is 1-periodic and the boundary condition $u(x + 1,t) = u(x,t)$ for all x, t; proving the existence and uniqueness of the solutions of such a problem. We use the semi-discrete approach together with the energy method.