  We study supersymmetric inhomogeneous field theories in 1+1 dimensions which
have explicit coordinate dependence. Although translation symmetry is broken,
part of supersymmetries can be maintained. In this paper, we consider the
simplest inhomogeneous theories with one real scalar field, which possess an
unbroken supersymmetry. The energy is bounded from below by the topological
charge which is not necessarily nonnegative definite. The bound is saturated if
the first-order Bogomolny equation is satisfied. Non-constant static
supersymmetric solutions above the vacuum involve in general a zero mode
although the system lacks translation invariance. We consider two inhomogeneous
theories obtained by deforming supersymmetric sine-Gordon theory and $\phi^6$
theory. They are deformed either by overall inhomogeneous rescaling of the
superpotential or by inhomogeneous deformation of the vacuum expectation value.
We construct explicitly the most general supersymmetric solutions and obtain
the BPS energy spectrum for arbitrary position-dependent deformations. Nature
of the solutions and their energies depend only on the boundary values of the
inhomogeneous functions. The vacuum of minimum energy is not necessarily a
constant configuration. In some cases, we find a one-parameter family of
degenerate solutions which include a non-vacuum constant solution as a special
case.
