a system of bilinear equations by introducing the same type of the dependent variable transformation as that employed in I for the bilinearization of the FL equation under vanishing boundary condition. We also demonstrate that this system yields a trilinear equation which will play a crucial role in our analysis.

The bilinearization of the FL equation (1.1) is established by the following proposition:

[**Proposition 2.1.**]{} [*By means of the dependent variable transformation $$u=\rho\,{\rm e}^{{\rm i}(\kappa x-\omega t)}\,{g\over f}, \eqno(2.1)$$ with $\omega=1/\kappa+2\rho^2$, equation (1.1) can be decoupled into the following system of bilinear equations for the tau functions $f$ and $g$ $$D_tf\cdot f^*-{\rm i}\rho^2(gg^*-ff^*)=0, \eqno(2.2)$$ $$D_xD_tf\cdot f^*-{\rm i}\rho^2D_xg\cdot g^*+{\rm i}\rho^2D_xf\cdot f^*+2\kappa\rho^2(gg^*-ff^*)=0, \eqno(2.3)$$ $$D_xD_tg\cdot f+{\rm i}\kappa D_tg\cdot f-{\rm i}\omega D_xg\cdot f=0. \eqno(2.4)$$ Here, $f=f(x, t)$ and $g=g(x, t)$ are complex-valued functions of $x$ and $t$, and the asterisk appended to $f$ and $g$ denotes complex conjugate and the bilinear operators $D_x$ and $D_t$ are defined by $$D_x^mD_t^nf\cdot g=\left({\partial\over\partial x}-{\partial\over\partial x^\prime}\right)^m
\left({\partial\over\partial t}-{\partial\over\partial t^\prime}\right)^n
f(x, t)g(x^\prime,t^\prime)\Big|_{ x^\prime=x,\,t^\prime=t},  \eqno(2.5)$$ where $m$ and $n$ are nonnegative integers.*]{}

[**Proof.**]{} Substituting (2.1) into (1.1) and rewriting the resultant equation in terms of the bilinear operators, equation (1.1) can be rewritten as $${1\over f^2}(D_xD_tg\cdot f+{\rm i}\kappa D_tg\cdot f-{\rm 