Kink localization under asymmetric double-well potentials

We study diffuse phase interfaces under asymmetric double-well potential energies with degenerate minima and demonstrate that the limiting sharp profile, for small interface energy cost, on a finite space interval is in general not symmetric and its position depends exclusively on the second derivatives of the potential energy at the two minima (phases). We discuss an application of the general result to porous media in the regime of solid-fluid segregation under an applied pressure and describe the interface between a fluid-rich and a fluid-poor phase.

Figure 1

Schematic description of the results in Theorems 1 and 2. The thick lines are the profiles at two finite values of $k$; the steepest one corresponds to the smallest value of $k$.

Figure 2

Schematic description of the heuristic interpretation of equation (7). The thick line is the profile at finite $k$, whereas the thin discontinuous line is its $k\to 0$ limit.

Figure 3

Kink profiles for the strain field $\varepsilon$ for the potential energy (11, 12, 13) with $k_1=k$, $k_2=k$, $k_3=k$, $k={10}^{-1},{10}^{-2},{10}^{-3}$, $\alpha =100$, $a=1/2$, and $b=1$.

Figure 4

Kink profiles for the fluid density field $m$ for the same parameters as in Fig. 3.

Figure 5

Kink profiles for the strain field $\varepsilon$ for the potential energy (11, 12, 13) with $k_1=k$, $k_2=k/2$, $k_3=k/3$, $k={10}^{-1},{10}^{-2},{10}^{-3}$, $\alpha =100$, $a=1/2$, and $b=1$.

Figure 6

Kink profiles for the fluid density field $m$ for the same parameters as in Fig. 5.

Figure 7

Graph of the kink for the potential (19) for ${\omega }_0=1$, $b_1=1$, $b_2=5$, and $u_0=0.5$. The profiles correspond to $k=1.0,0.5,0.25,0.1$ in increasing order of steepness.

Figure 8

Graph of the kink for the potential (20) for $a=1$. The profiles correspond to $k=3.0,1.5,1.0,0.5,0.45$ in increasing order of steepness. The two profiles corresponding to $k=0.5$ and $k=0.45$ are almost coincident in the picture.