Effective integration of the Nonlinear Vector Schr\"odinger Equation

TL;DR

This paper develops an explicit algebro-geometric method to solve the two-component Nonlinear Vector Schr"odinger equation (Manakov system) using theta-functions, enabling practical calculations for applied problems.

Contribution

It provides a comprehensive, explicit solution framework for the Manakov system using spectral curves and theta-functions, making the solutions computationally accessible.

Findings

Explicit solutions in terms of theta-functions are derived.

Solutions are expressed with algebraic constants and prime-forms for direct computation.

Simplest solutions are discussed as special cases.

Abstract

A comprehensive algebro-geometric integration of the two component Nonlinear Vector Schr\"odinger equation (Manakov system) is developed. The allied spectral variety is a trigonal Riemann surface, which is described explicitly and the solutions of the equations are given in terms of theta-functions of the surface. The final formulae are effective in that sense that all entries like transcendental constants in exponentials, winding vectors etc. are expressed in terms of prime-form of the curve and well algorithmized operations on them. That made the result available for direct calculations in applied problems implementing the Manakov system. The simplest solutions in Jacobian theta-functions are given as particular case of general formulae and discussed in details.