Greediness isn't an issue here, because the character class [^\$] doesn't allow a $ to be part of the captured fragments.

Alternatively, you could use a non-greedy pattern: /\$(.+?)\$/g

my $latex = q|We find that the generical scale behavior of structure f
+unctions in the inertial range is not self-similar $S_n(\ell)\propto 
+\ell^{\zeta_n}$ but includes an \u201cexponential self-similar\u201d 
+behavior $S_n(\ell) \propto \exp[\zeta_n\alpha^{-1} \ell^{\alpha}]$ w
+here $\alpha$ is a parameter proportional to the inverse of the logar
+ithm of the Reynolds number.|;

my @equations = $latex =~ /\$(.+?)\$/g;

say for @equations;
[download]

Output:

S_n(\ell)\propto \ell^{\zeta_n}
S_n(\ell) \propto \exp[\zeta_n\alpha^{-1} \ell^{\alpha}]
\alpha
[download]

Thank you, Eliya. That code is clear and now my problem is solved.