content_image/333881.jpg
Taking the initial brane to be located at y = 0, we find solutions to the initial-boundary value problem at this brane with Dirichlet boundary conditions imposed on the field φ . Examples of nodeless solutions to the initial value problem are shown in Fig. 4. To ensure that φ obeys Dirichlet boundary conditions on the second brane, we locate the second brane at the first point (other than y = 0) where the profile of φ vanishes (the vertical dashed lines in Fig. 4). 2 Once the position of the second brane is identified, the final-boundary conditions are determined from (43) and (44). By varying the initial shooting conditions, φ ′ 1 ≡ φ $^{′}$( y$_{1}$ ) and χ ′ 1 ≡ χ $^{′}$( y$_{1}$ ), and repeating this process of finding solutions, identifying the location of the second brane, and determining the final-boundary conditions, we generate level-curves of µ 2 and Σ$_{2}$. These are plotted in Fig. 5. Notice that most µ 2 contours cross each Σ$_{2}$ just once, signifying that there is a single solution for the corresponding action with the kink potential of Eq. (45). However, some contours cross each other more than once (see, for example, the circles in Fig. 5.) Furthermore there is only a finite region in the φ ′ $_{1}$- χ ′ 1 parameter space where solutions exist. If either | φ ′ 1 | or | χ ′ 1 | are increased sufficiently, the solution to the initial value problem blows up. In that case the boundary value problem has no solution, since a second boundary where φ = 0 does not exist. It is therefore possible to scan the entire allowed φ ′ $_{1}$- χ ′ 1 space and examine whether multiple solutions with the same action exist. 

C. Example 2: Higher-Order Potential 

In this second example we take a slightly more complicated kink potential 

<img src='content_image/333881.jpg'>

The other potentials and boundary conditions are the same as in the previous example, the only difference being the dynamical evolution of the system due to the new potential V ( φ ). We choose this potential because, contrary to the potential in our first example, in the limit of weak gravity and flat spacetime, it leads to multiple solutions to the same boundary value problem [40, 41] due to the nonlinear nature of the equations. 

