Lax pair and vector solitons for a variable-coefficient coherently-coupled nonlinear Schrödinger system in the nonlinear birefringent optical fiber

Nowadays, with respect to the nonlinear birefringent optical fibers, efforts have been put into investigating the coupled nonlinear Schrödinger (NLS) systems. In this paper, symbolic computation on a variable-coefficient coherently-coupled NLS system with the alternate signs of nonlinearities is performed. Under a variable-coefficient constraint Ω(t)2=[lnγ(t)]t2-[lnγ(t)]tt$ \Omega (t)^{2}=[\ln \gamma (t)]_{t}^{2}-[\ln \gamma (t)]_{tt} $, the system is shown to be integrable in the Lax sense with a Lax pair constructed, where t is the normalized time, γ(t)$ \gamma (t) $ is the strength of the four wave mixing terms, and Ω(t)2$ \Omega (t)^{2} $ is the strength of the anti-trapping parabolic potential. With an auxiliary function, bilinear forms, vector one- and two-soliton solutions are obtained. Figures are displayed to help us study the vector solitons: When γ(t)$ \gamma (t) $ is a constant, vector soliton propagates stably with the amplitude and velocity unvarying (vector soliton’s amplitude changes with the change of that constant, while its velocity can not be affected by that constant); When γ(t)$ \gamma (t) $ is a t-varying function, i.e. Ω(t)2≠0$ \Omega (t)^{2}\ne 0 $, amplitude and velocity of the vector soliton both vary with t increasing, while Ω(t)2$ \Omega (t)^{2} $ affects the vector soliton’s amplitude and velocity. With the different γ(t)$ \gamma (t) $ or Ω(t)2$ \Omega (t)^{2} $, interactions between the amplitude- and velocity-unvarying vector two solitons and those between the amplitude- and velocity-varying vector two solitons are displayed, respectively. By virtue of the system and its complex-conjugate system, conservation laws for the vector solitons, including the total energy and momentum, are constructed.