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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.

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

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