Example 14 Let positive integers $a, m, n$ satisfy $\sqrt{a^{2}-4 \sqrt{2}}=\sqrt{m}-\sqrt{n}$. Then the number of such values of $a, m, n$ is ( ).
(A) one set
(B) two sets
(C) more than two sets
(D) does not exist
Solving: Squaring both sides of the original equation, we get
$$
a^{2}-4 \sqrt{2}=m+n-2 \sqrt{m n} \text {. }
$$

Since $a$, $m$, and $n$ are positive integers, $a^{2}-4 \sqrt{2}$ and $\sqrt{m n}$ are irrational numbers. Therefore,
$$
\left\{\begin{array}{l}
\sqrt{m n}=\sqrt{8}, \\
m+n=a^{2},
\end{array}\right.
$$

Thus, $\left\{\begin{array}{l}m n=8, \\ m+n=a^{2} .\end{array}\right.$
Given that $m>n$, the only solution is $m=8$, $n=1$, and $a=3$. Therefore, the answer is (A).