We therefore consider the 5D action for gravity and N free scalar fields where M$_{∗}$ ≡ (8 πG ) − 1 / $^{3}$, G is the 5D Newton’s constant, R is the 5D Ricci scalar, and Λ is the 5D cosmological constant. The full scalar potential in the bulk is W ( χ$_{a}$ ), and the brane potentials are λ$_{i}$ ( χ$_{a}$ ). As before, we take the 5D line element of the form where γ$_{µν}$ is the induced metric on the 4D hypersurfaces of constant y , which foliate the extra dimension. The 5D Einstein and field equations are where $^{(4)}$R is the 4D Ricci scalar associated with the induced 4D metric γ$_{µν}$ , which we have left arbitrary. The boundary conditions for the system are determined by Israel junction conditions at each brane. These are obtained by integrating the equations of motion over an infinitesimally small interval across each brane, giving These yield N conditions on each brane, which is exactly the number of data that need to be specified in order for equations (32) and (34) to form a well-posed problem. Note that the above boundary value problem consists of a system of coupled nonlinear differential equations. Finding solutions analytically for such a setup is highly unlikely, although it is still possible to proceed in the opposite direction, i.e. given a particular analytical solution one can obtain the setup from which it originates. To do so, one relies 13 [ σ $^{′}$]$_{y}$$_{i}$ ≡ lim ϵ → $_{0}$[ σ $^{′}$( y$_{i}$ + ϵ ) − σ $^{′}$( y$_{i}$ − ϵ )] = 1 3 M 3 ∗ λ$_{i}$ ( χ$_{a}$ ) | y i (35) [ χ ′ $_{a}$]$_{y}$$_{i}$ ≡ lim ϵ → $_{0}$[ χ ′ $_{a}$( y$_{i}$ + ϵ ) − χ ′ $_{a}$( y$_{i}$ − ϵ )] = ∂λ$_{i}$ ∂χ$_{a}$ ∣ y i . (36) χ ′′ a − 4 σ $^{′}$χ′ a − ∂W ∂χ$_{a}$ − ∑ i =1 , 2 ∂λ$_{i}$ ∂χ$_{a}$ δ ( y − y$_{i}$ ) = 0 , (34) σ ′ 2 − Λ 6 + $^{(4)}$R 12 e 2 σ = 1 6 M 3 ∗ ( N ∑ a 1 2 χ ′ 2 a − W ( χ$_{a}$ ) ) (33) σ ′′ − σ ′ 2 + Λ 6 = 1 2 M 3 ∗ ( N ∑ a 1 2 χ ′ 2 a + 1 3 W ( χ$_{a}$ ) + 2 3 ∑ i =1 , 2 λ$_{i}$ ( χ$_{a}$ ) δ ( y − y$_{i}$ ) ) (32) ds 2 = e − 2 σ ( y $^{)}$γ$_{µν}$ ( x ) dx $^{µ}$dxν − dy 2 , (31) S = − M 3 ∗ 2 ∫ d 5 $_{x}$√ − g [ R − 2Λ] + ∫ d 5 $_{x}$√ − g [ N ∑ a =1 1 2 g $^{MN}$( ∂$_{M}$ χ$_{a}$ )( ∂$_{N}$ χ$_{a}$ ) − W ( χ$_{a}$ ) − ∑ i =1 , 2 λ$_{i}$ ( χ$_{a}$ ) δ ( y − y$_{i}$ ) ] , (30)