Boundedness at the Riemann-constants shift for the finite-gap Boussinesq solution

Prove that, when the constant vector C in the finite-gap solution is chosen to be the vector of Riemann constants u[K], the finite-gap Boussinesq solution w(x,t) = −3 ℘_{1,1}(u + C) and v(x,t) = 2 ℘_{1,1,1}(u + C) − (3/2) ℘_{1,2}(u + C) (with u = (x, t, 0, …)^t) is bounded for all x,t.

Background

The explicit finite-gap solution obtained in the paper expresses the Boussinesq fields in terms of Kleinian ℘-functions associated with the (3,3N+1)-curve. Numerical exploration shows solutions with singularities for various shifts C in the Jacobian.

The authors conjecture that taking C equal to the vector of Riemann constants u[K] yields bounded solutions. Establishing boundedness at this canonical shift would provide a constructive route to non-singular finite-gap solutions and clarify the role of the half-period structure in controlling poles of ℘-functions.