Science and Mathematics Faculty Publications
Document Type
Article
Publication Date
5-2016
Journal Title
Physica D: Nonlinear Phenomena
Volume
321-322
First Page
16
Last Page
38
DOI
http://dx.doi.org/10.1016/j.physd.2016.03.002
Abstract
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.
Keywords
Nonlinear Schrödinger equation; ultra-elliptic solutions; two-phase solutions
Recommended Citation
Wright, Otis C. III, "Effective Integration of Ultra-Elliptic Solutions of the Focusing Nonlinear Schrödinger Equation" (2016). Science and Mathematics Faculty Publications. 326.
https://digitalcommons.cedarville.edu/science_and_mathematics_publications/326
