For a function $q(t,x,y)$, what does $q^*$, $q_x^*$ mean?

I am having some trouble understanding the notation here. The original text is not in English so I doubt I can give references here to be useful, but it is about non-linear wave motion equations. It is specifically called the "Davey-Stewartson type I" equation (DSI) which I am working on for a project.

However, I don't understand what the following $q^*$ means. to explain, $L_1,L_2$ are operators, which is used to express the DSI as a Lax pair $[L_1,L_2]=0$ where each $J,Q,W,D$ are given as shown. The original DSI equation is

$$iq_t+q_{xx}+q_{yy}+(U+V)q=0$$

where

$$U+\int|q|_x^2dy+u(x,t), V=\int|q|_y^2dx+v(x,t)$$

for some $u,v$. The idea is apparently to solve for $q$, and I don't understand what $q^*$ is in the diagram above, say, in $Q,W$. The paper does not tell me, or define what the notation is supposed to mean.

I thought to the experienced eye, it might be obvious what they are talking about with $q^*$. Does anyone know? Or have some guesses as to what probably it is?

• Might be a complex conjugate of $q$. If $q : \mathbb{R}^3 \mapsto \mathbb{C}$. – echzhen Feb 27 '16 at 17:36