Briefly: I have two Lax pairs in matrix form and using compatibility condition I have found a nonlinear partial differential equation system. I have searched this system very much seems the nonlinear Schrodinger Equation system (in wikipedia) but I am stuck how to derive it nonlinear Schrodinger eq. from these two nonlinear pde, I have tried some transformations but could not do it.
Work: Lax Pairs $$X=\begin{bmatrix} -\lambda & u\\ v & \lambda \\ \end{bmatrix}\quad,\quad T=\begin{bmatrix} -\lambda^2+\frac12uv ,&\lambda u-\frac12u_x \\ \lambda v+\frac12v_x & \lambda^2 -\frac12uv \\ \end{bmatrix}\tag1$$
Compatibility condition is $$D_tX-D_xT+[X,T]=0$$ where $$[X,T]=XT-TX$$ $$\begin{bmatrix} 0 & u_t+\frac12u_{xx}-u^2v\\ v_t-\frac12v_{xx}+uv^2 & 0\\ \end{bmatrix}=\begin{bmatrix} 0 & 0\\ 0 & 0 \\ \end{bmatrix}$$
so both have to be zero as following
$$u_t+\frac12u_{xx}-u^2v=0\\ v_t-\frac12v_{xx}+uv^2=0\tag2$$
suggested transformations are $u=\pm v^*$ but what does it mean? and my system $(2)$ has no $i$ coefficients. So I am stuck here how to derive nonlinear Schrodinger equation namely $$i\Phi_t+\Phi_{xx}+\sigma \Phi^2\Phi^*=0$$ from this system? I am sure about it because it is an exercise problem and I cannot figure it out.