Effective integration of ultra-elliptic solutions of the focusing nonlinear Schrödinger equation

• Kleinian ultra-elliptic function formulas for two-phase solutions of the focusing NLSE.
• Theta function formulas for two-phase solutions with explicit construction of parameters.
• Real loci of the Dirichlet eigenvalues parametrized by wave amplitude and wavenumber.
• Simple formulas for the extrema of the modulus of the two-phase solution.

An effective integration method based on the classical solution of the Jacobi inversion problem, using Kleinian ultra-elliptic functions and Riemann theta functions, is presented for the quasi-periodic two-phase solutions of the focusing cubic nonlinear Schrödinger equation. Each two-phase solution with real quasi-periods forms a two-real-dimensional torus, modulo a circle of complex-phase factors, expressed as a ratio of theta functions associated with the Riemann surface of the invariant spectral curve. The initial conditions of the Dirichlet eigenvalues satisfy reality conditions which are explicitly parametrized by two physically-meaningful real variables: the squared modulus and a scalar multiple of the wavenumber. Simple new formulas for the maximum modulus and the minimum modulus are obtained in terms of the imaginary parts of the branch points of the Riemann surface.