Left- (black line) and right- (red line) hand sides of Eq. (19) as a function of s/L, for three different values of l_t. Top panel: system is linearly stable with one solution (see circle). Bottom panel: system relaxes to a new equilibrium at a lower height (see the two complementary solutions) and the trivial solution becomes an unstable solution. In all cases, L = 10 H and ρ_t = 100 ρ₀; l_t = L/110 (top panel), l_t = l_tb (middle panel), and l_t = L/30 (bottom panel). The value l_tb ≈ L/77.08 is obtained when the derivatives of the two curves at s = L/2 are the same (see Eq. (20)). The vertical dashed lines represent the edges of the domain of the variable s, namely s = l_t/2 and s = L − l_t/2.