Verification of genericity, density

Finally, [83

, Theorem 2, p. 1190] yields the conclusion that $𝒢$ is dense:

Theorem 3 $𝒢$ and $𝒢c$ are dense in $𝒮p$ and $𝒮p,c$ , respectively, with respect to the $C ∞$ -topology on initial data.

Here $𝒮p$ and $𝒮p,c$ are defined in [83 , Definition 1, p. 1188]:

Definition 8 Let $𝒮p$ denote the set of smooth solutions to Equations (16 )–(17 ) on $ℝ × S1$ , and let $𝒮 p,c$ denote the subset of $𝒮 p$ obeying