Maximal Densities of Finite-Gap Solutions of the Sine-Gordon Equation

We establish a sharp upper bound on the densities of finite-gap solutions of the sine-Gordon equation. The bound is derived directly from the finite-dimensional hierarchy, without explicit integration of the finite-gap solutions. The maximal density is determined by the roots of the invariant polynomial. An analogous sharp upper bound is established for a bounded class of finite-gap solutions of the sinh-Gordon equation.