 96357 Rothos V.M.
 Study of a System of Coupled NLS Equations
Under Periodic Boundary Conditions
(30K, LaTeX)
Aug 10, 96

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Abstract. The theory of a simple NLS equation under
periodic boundary conditions has been recently studied by Y.Li et al, using the
Floquet spectral analysis of its
associated ZakharovShabat linear operator, with the purpose of describing
homoclinic solutions of perturbations
of the NLS equation. Here, we show that a system of three coupled NLS
equations
$$iq_{_{1t}}=q_{_{1xx}}+2{\vert q_{_{1}} \vert}^2q_{_{1}}+4q_{_{1}}{\vert q_{_{2}} \vert}^2
+2q^{2}_{_{2}}q^{*}_{_{3}}$$
$$iq_{_{2t}}=q_{_{2xx}}+2{\vert q_{_{1}} \vert}^2q_{_{2}}+2{\vert q_{_{2}} \vert}^2q_{_{2}}
+2{\vert q_{_{3}} \vert}^2q_{_{2}}+2q_{_{1}}q^{*}_{_{2}}q_{_{3}}$$
$$iq_{_{3t}}=q_{_{3xx}}+2{\vert q_{_{3}} \vert}^2q_{_{3}}+4q_{_{3}}{\vert q_{_{2}} \vert}^2
+2q^{2}_{_{2}}q^{*}_{_{1}}$$
with periodic boundary conditions \( q_{_{i}}(x+l)=q_{_{i}}(x)\quad
{i=1,2,3} \) is integrable and specify the corresponding
ZakharovShabat linear operator. We also attempt to describe its homoclinic orbits
and give new results for its associated nonintegrable perturbations
through a Mel'nikov approach, using the spectral analysis of Lax 's operator
and Floquet theory.
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