96-357 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 Zakharov--Shabat 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 Zakharov--Shabat linear operator. We also attempt to describe its homoclinic orbits and give new results for its associated non--integrable perturbations through a Mel'nikov approach, using the spectral analysis of Lax 's operator and Floquet theory.

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