Time evolution of the scattering data for a fourth-order linear differential operator

The time evolution of the scattering and spectral data is obtained for the differential operator ⁴/dx⁴ +/dx u(x,t)/dx+v(x,t), where $u(x,t)$ and $v(x,t)$ are real-valued potentials decaying exponentially as x→±∞ at each fixed $t.$ The result is relevant in a crucial step of the inverse scattering transform method that is used in solving the initial-value problem for a pair of coupled nonlinear partial differential equations satisfied by $u(x,t)$ and $v(x,t).$