Bilinear approaches for a finite-dimensional Hamiltonian system

The paper investigates a finite-dimensional Hamiltonian system through bilinear approaches, i.e. the Hirota method and Wronskian technique. This system is obtained by constraining the potential of the sine-Gordon equation with a symmetry. We show that the potential can be a summation of squares of wave functions and these wave functions can be precisely described as Wronskians. Furthermore, our work also demonstrates that the bilinear approaches are efficient tools not only for partial differential equations but also for ordinary differential equations.