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