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

Abstract

The time evolution of the scattering and spectral data is obtained for the differential operator $\frac{d^4}{d{x}^4}+\frac{d}{dx}u(x,t)\frac{d}{dx}+v(x,t),$ where $u(x,t)$ and $v(x,t)$ are real-valued potentials decaying exponentially as $x\to \pm \infty$ 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 $v(x,t).$

19 pages

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