Periodic solutions of a derivative nonlinear Schrödinger equation: Elliptic integrals of the third kind

Authors:
K. W. Chow;T. W. Ng
Affiliations:
Department of Mechanical Engineering, University of Hong Kong, Pokfulam, Hong Kong;Department of Mathematics, University of Hong Kong, Pokfulam, Hong Kong
Venue:
Journal of Computational and Applied Mathematics
Year:
2011

Citing 3
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Precise and fast computation of a general incomplete elliptic integral of third kind by half and double argument transformations

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Abstract

The nonlinear Schrodinger equation (NLSE) is an important model for wave packet dynamics in hydrodynamics, optics, plasma physics and many other physical disciplines. The 'derivative' NLSE family usually arises when further nonlinear effects must be incorporated. The periodic solutions of one such member, the Chen-Lee-Liu equation, are studied. More precisely, the complex envelope is separated into the absolute value and the phase. The absolute value is solved in terms of a polynomial in elliptic functions while the phase is expressed in terms of elliptic integrals of the third kind. The exact periodicity condition will imply that only a countable set of elliptic function moduli is allowed. This feature contrasts sharply with other periodic solutions of envelope equations, where a continuous range of elliptic function moduli is permitted.