we shall see now. To this end, following the procedure used in \[11, 12\], we introduce the gauge transformation $$q=v\,{\rm exp}\left[{\rm i}\int_{-\infty}^x(\rho^2-|v|^2)dx\right], \eqno(3.31a)$$ as well as the dependent variable transformation for $v$ $$v=\rho\,{\rm e}^{{\rm i}(\kappa x-\omega^\prime t)}\,{h\over f}. \eqno(3.31b)$$ Then, equation (3.27) can be decoupled to the system of bilinear equations for $f$ and $h$ $$D_xf\cdot f^*-{\rm i}\rho^2(hh^*-ff^*)=0, \eqno(3.32)$$ $$D_x^2f\cdot f^*-{\rm i}\rho^2D_xh\cdot h^* +\rho^2(2\kappa+\rho^2)(hh^*-ff^*)=0, \eqno(3.33)$$ $${\rm i} D_th\cdot f+2{\rm i}(\kappa+\rho^2)D_xh\cdot f+D_x^2h\cdot f=0. \eqno(3.34)$$ In view of (3.32), the modulus of $v$ is given in terms of the tau function $f$ by $$|v|^2=\rho^2+{\rm i}\,{\partial \over \partial x}\,{\rm ln}\,{f^*\over f}, \eqno(3.35)$$ which, combined with (3.31), yields the formula (3.29). Note from (3.31a) that $|q|^2=|v|^2$. It may be checked by direct computation that the tau functions $f$ and $h$ from (3.29b) with (3.30) satisfy the above bilinear equations.

It is important to realize that we can take the limit $\kappa\rightarrow 0$ for the solution (3.29) since the dispersion relation is not singular at $\kappa=0$. This gives the $N$-soliton solution of the derivative NLS equation on a constant background which has been studied extensively using various exact methods of solution such as the IST \[13-16\], Bäcklund transformation \[17, 18\] and Hirota’s direct method \[19\]. 