i}\omega D_xg\cdot f)$$ $$-{g\over f^3f^*}\bigl\{f^*D_xD_tf\cdot f-2\kappa\rho^2f^2f^*-2{\rm i}\rho^2g^*(g_xf-gf_x+{\rm i}\kappa fg)\bigr\}=0. \eqno(2.6)$$ Inserting the identity $$f^*D_xD_tf\cdot f=fD_xD_tf\cdot f^*-2f_xD_tf\cdot f^*+f(D_tf\cdot f^*)_x, \eqno(2.7)$$ which can be verified by direct calculation, into the second term on the left-hand side of (2.6), one modifies it in the form $${1\over f^2}(D_xD_tg\cdot f+{\rm i}\kappa D_tg\cdot f-{\rm i}\omega D_xg\cdot f)$$ $$-{g\over f^3f^*}\Bigl[f\bigl\{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^*)\bigr\}$$ $$-2f_x\bigl\{D_tf\cdot f^*-{\rm i}\rho^2(gg^*-ff^*)\bigr\}+f\bigl\{D_tf\cdot f^*-{\rm i}\rho^2(gg^*-ff^*)\bigr\}_x\Bigr]=0. \eqno(2.8)$$ By virtue of equations (2.2)-(2.4), the left-hand side of (2.8) vanishes identically. $\Box$

It follows from (2.1) and (2.2) that $$|u|^2=\rho^2+{\rm i}\,{\partial \over \partial t}\,{\rm ln}\,{f^*\over f}. \eqno(2.9)$$ The above formula gives the modulus of $u$ in terms of the tau function $f$.

[**Proposition 2.2.**]{} [*The [*trilinear*]{} equation for $f$ and $g$ $$f^*\left\{g_{xt}f-(f_x-{\rm i}\kappa f)g_t-{\rm i}\left({1\over \kappa}+\rho^2\right)(g_xf-gf_x)\right\}
=f_t^*(g_xf-gf_x+{\rm i}\kappa fg), \eqno(2.10)$$ is a consequence of the bilinear equations (2.2)-(2.4).*]{}

[**Proof.**]{} By direct calculation, one can show the following trilinear identity among the tau functions $f$ and $g$: $$f^*\left\{g_{xt}f-(f_x-{\rm i}\kappa f)g_t-{\rm i}\left({1\over \kappa}+\rho^2\right)(g_xf-gf_x)\right\}
-f_t^*(g_xf-gf_x+{\rm i}\kappa fg)$$ $$=f^*(D_xD_tg\cdot f+{\rm i}\kappa D_tg\cdot f-{\rm i}\omega D_xg\cdot f)$$ $$-{g\over 2}\Bigl[\bigl\{D_tf\cdot f^*-{\rm i}\rho^2(gg^*-ff^*)\bigr\}_x+(D_xD_tf\cdot f^*-{\rm i}\rho^2D_xg\cdot g^*+{\rm i}\rho^2D_xf\cdot f^*-2{\rm i}\kappa D_tf\cdot f^*)\Bigr]$$ $$+g_x\bigl\{D_tf\cdot f^*-{\rm i}\rho^2(gg^*-ff^*)\bigr\}. \eqno(2.11)$$ Replacing a term $2{\rm i}\kappa D_tf\cdot f^*$ on the right-hand side of (2.11) by (2.2), the right-hand 