Jacobi elliptic solutions of lambda phi(4) theory in a finite domain

The general static solutions of the scalar field equation for the potential V(phi) = -1/2 M-2 phi(2) + lambda/4 phi(4) are determined for a finite dom ain in (1 + 1)-dimensional space-time. A family of real solutions is descri bed in terms of Jacobi Elliptic Functions. We show that the vacuum-vacuum b oundary conditions can be reached by elliptic cn-type solutions in a finite domain, such as that of the Kink, for which they are imposed at infinity. We prove uniqueness for elliptic sn-type solutions satisfying Dirichlet bou ndary conditions in a finite interval (box) as well the existence of a mini mal mass corresponding to these solutions in a box. We defined expressions for the "topological charge," "total energy" (or cla ssical mass) and "energy-density" for elliptic sn-type solutions in a finit e domain. For large length of the box the conserved charge, classical mass and energy density of the Kink are recovered. Also, we have shown that usin g periodic boundary conditions the results are the same as in the case of D irichlet boundary conditions. In the case of antiperiodic boundary conditio ns all elliptic sn-type solutions are allowed.