content_image/373257.jpg content_image/373258.jpg content_image/373259.jpg content_image/373260.jpg
one of the scalar fields in terms of the others (see Appendix A for details). The resulting N independent equations are 

<img src='content_image/373257.jpg'>

where H ≡ a $_{a}$, $^{(4)}$□ ≡ γ $^{µν}$∂$_{µ}$ ∂$_{ν}$ , and in equation (51), as throughout, we have assumed that there are no direct couplings between the 5D scalar fields in the scalar potential (in Appendix A we derive the general form of these equations when couplings between the fields are included). These dynamical equations can be written more compactly as 

<img src='content_image/373258.jpg'>

where ξ has suppressed discrete indices which run over the N − 1 fundamental scalar fields, Ψ is the graviscalar, the D y i are y -dependent differential operators (i.e., linear differential oper- ators having y -dependent coefficients and acting only on functions of y ) also with suppressed discrete indices, and □ ≡ γ $^{µν}$∂$_{µ}$ ∂$_{ν}$ is the 4D wave operator. 

The boundary conditions are determined by integrating the equations of motion across each brane. Integrating equations (50) and (51), these are found to be 

<img src='content_image/373259.jpg'>

These boundary conditions can be put in the form 

<img src='content_image/373260.jpg'>

