KdV Schrodinger spectral problem
= KdV Schrodinger spectral problem
{c}
{title2=$-\phi_{xx}+u\phi=\lambda\phi$}
The inverse-scattering spectral problem associated with this KdV convention is
$$
-\phi_{xx}+u(x,t)\phi=\lambda(t)\phi.
$$
For
$$
Q=\phi_t+u_x\phi-2(u+2\lambda)\phi_x,
$$
differentiation gives the <Wronskian> identity
$$
\partial_x(\phi_xQ-\phi Q_x)
=\phi^2\bigl[\dot\lambda-(u_t+u_{xxx}-6uu_x)\bigr].
$$