Nonmonotonic solutions of the Falkner-Skan boundary layer equation

We investigate the equation $f”’ + ff” + \beta (1 - f’^2) = 0$ together with boundary conditions $f\left ( 0 \right ) = f’\left ( 0 \right ) = 0$ and $f’\left ( \infty \right ) = 1$. Here $\beta$ is negative. Previous results are summarized which describe solutions which satisfy $\left | f’ \right | < 1$ for all $\eta \ge 0$. It is shown that there is a sequence $\left \{ \beta _j \right \}_{j \in \mathbb {N}}$ of decreasing, negative values of $\beta$, and a corresponding sequence $\left \{ f_j \right \}_{j \in \mathbb {N}}$ of solutions such that for each $j \in \mathbb {N}$ the equation $f_j’ - 1 = 0$ has exactly $j$ positive solutions and for some $\mu _j > 0, f_j’ = 1 + o\left ( \exp \left ( - \mu _j\eta \right ) \right )$ as $\eta \to \infty$.