GLYPH<B9>GLYPH<BE> GLYPH<B9>GLYPH<BD>GLYPH<BA>GLYPH<BH> GLYPH<B9>GLYPH<BD> GLYPH<B9>GLYPH<BC>GLYPH<BA>GLYPH<BH> GLYPH<BC> GLYPH<BC>GLYPH<BA>GLYPH<BH> GLYPH<BC> GLYPH<BD> GLYPH<BE> GLYPH<BF> GLYPH<BG> GLYPH<BH> [ M 3 / 2 ∗ ] GLYPH<B4>GLYPH<BT>GLYPH<B5> y [ M − 1 ∗ ] φ ( y ) χ ( y ) σ ( y ) GLYPH<BD> GLYPH<BE> GLYPH<BF> GLYPH<BG> GLYPH<BH> GLYPH<B4>GLYPH<BU>GLYPH<B5> φ ( y ) χ ( y ) σ ( y ) FIG. 6: Profiles of the scalar backgrounds φ ( y ) and χ ( y ) as well as the warp factor σ ( y ), showing the two possible solutions (panels A and B) to the same boundary value problem defined by the physical parameters m 2 χ = − 0 . 5 M 2 $_{∗}$, µ 2 1 = − 0 . 25 M 2 $_{∗}$, µ 2 = − 5 M 2 $_{∗}$, Σ$_{1}$ = − 2 M 4 $_{∗}$, Σ$_{2}$ = 0 . 56 M 4 $_{∗}$, m 2 φ = 0 . 5 M 2 $_{∗}$, λ = 2 M − 1 ∗ , ξ = 6 M − 4 ∗ , and Λ = 0. can be determined analytically. Unfortunately, in the case of warped extra dimensions, the question of stability is complicated by the presence of multiple scalar fields and their coupled dynamics. Here we begin to study the perturbative stability of these kinked configurations. We derive the linearized equations and reformulate the problem in terms of a matrix Sturm- Liouville problem. However, the full analysis requires matrix Sturm-Liouville methods which we omit and leave for future work. We begin by expanding the metric to first-order. Instead of the coordinates in (7), in this section it will be more convenient to choose coordinates so that the metric takes the form ds 2 = a $^{2}$( y ) ( γ$_{µν}$ ( x ) dx $^{µ}$dxν − dy $^{2}$) . (47) 20