Bounded real-valued solutions and relation to known Boussinesq solutions

Develop bounded and real-valued solutions of the Boussinesq equation within the finite-gap framework based on (3,3N+1)-curves, and ascertain their connection with known solutions of the Boussinesq equation.

Background

While the paper derives explicit finite-gap solutions and investigates reality conditions, the authors report that only solutions with singularities were obtained in their computations. They highlight that achieving bounded, real-valued solutions remains unresolved.

Clarifying how such finite-gap solutions relate to existing classes of solutions (e.g., solitonic, breather-type, or solutions from inverse scattering/dressing methods) would integrate the algebro-geometric results into the broader solution theory for the Boussinesq equation.