Effective integration of the nonlinear vector Schrödinger equation

A comprehensive algebro-geometric integration of the two component Nonlinear Vector Schrödinger 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 θ -functions of th...

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Bibliographic Details
Published in:Physica. D 2007-01, Vol.225 (2), p.127-152
Main Authors: Elgin, J.N., Enolski, V.Z., Its, A.R.
Format: Article
Language:English
Subjects:
Exact sciences and technology
Finite gap solutions
Physics
Theta functions
Trigonal curve
Vector NLS equation
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Summary:A comprehensive algebro-geometric integration of the two component Nonlinear Vector Schrödinger 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 θ -functions of the surface. The final formulae are effective in the sense that all entries, like transcendental constants in exponentials, winding vectors etc., are expressed in terms of the 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 ϑ -functions are given as a particular case of general formulae and are discussed in detail.
ISSN:0167-2789
1872-8022
DOI:10.1016/j.physd.2006.10.005