Wronskian and Boundary condition

Here is a partial differential equation, which is KdV equation, $$u_t-6uu_x+u_{xxx}=0$$ with boundary conditions $$\psi\rightarrow e^{ikx},\quad \bar{\psi}\rightarrow e^{-ikx},\quad x\rightarrow\infty\\ \phi\rightarrow e^{-ikx},\quad \bar{\phi}\rightarrow e^{ikx}, \quad x\rightarrow-\infty$$

My problem is someone gave me a claim that Wronskian of one of the solutions and its conjugate is $$2ik$$, i.e., $$W(\psi,\bar{\psi})=2ik=-W(\psi,\bar{\psi})$$ Is it reasonable determining Wronskian by boundary condition?

It's important since if the Wronskian is nonzero, I can represent a solution by another solution with transmission/reflection coefficient(related to $$a(k)$$, $$b(k)$$). $$\phi(x,k)=a(k)\bar{\psi}(x,k)+b(k)\psi(x,k)\\ \bar{\phi}(x,k)=-\bar{a}(k)\psi(x,k)+\bar{b}(k)\bar{\psi}(x,k)$$

And here $$a(k)=\frac{W(\phi,\psi)}{2ik}$$, $$b(k)=-\frac{W(\phi,\bar{\psi})}{2ik}$$. I'm also not sure why the coefficients is in this way....

The claim comes from ''Ablowitz. Solitons, Nonlinear Evolution Equations and Inverse scattering.

• Please provide more context. Where does this problem come from? Commented Jan 23, 2022 at 11:58
• I append the source of this problem. Perhaps it's too hard for me to clarify whole frame. Thans for your advice. Commented Jan 23, 2022 at 12:30
• I'm still confused. What do $\phi,\psi$ have to do with $u$ ? Commented Jan 23, 2022 at 12:47
• $\phi$, $\psi$ are solutions of KdV equation under the boundary condition. Commented Jan 23, 2022 at 12:56