Relations of al Functions over Subvarieties in a Hyperelliptic Jacobian

The sine-Gordon equation has hyperelliptic al function solutions over a hyperelliptic Jacobian for $y^2=f(x)$ of arbitrary genus $g$. This article gives an extension of the sine-Gordon equation to that over subvarieties of the hyperelliptic Jacobian. We also obtain the condition that the sine-Gordon equation in a proper subvariety of the Jacobian is defined.