KdV Schrodinger spectral problem ($-{\varphi }_{xx}+u\varphi =\lambda \varphi$)

The inverse-scattering spectral problem associated with this KdV convention isFordifferentiation gives the Wronskian identity

A Jost solution of a one-dimensional Schrodinger equation is specified by plane-wave asymptotics at one spatial infinity. Comparing the left and right Jost solutions defines transmission and reflection coefficients.

For a rapidly decaying KdV potential, the scattering data consist of the continuous reflection coefficient, the negative discrete eigenvalues $-{\kappa }_n^2$, and a positive norming constant for each bound state.

For $u_t-6u{u}_x+u_{xxx}=0$, the discrete eigenvalues are fixed, whilewhen $c_n$ denotes the bound-state asymptotic amplitude. The squared amplitudes appearing in the Marchenko kernel consequently evolve by $e^{8{\kappa }_n^{3}t}$.

For a weak potential $u=ϵq$, the first Born approximation in the KdV Schrödinger convention is

For a rapidly decreasing KdV potential and a normalized bound-state eigenfunction, integration of the Wronskian identity over the real line gives ${\dot{\lambda }}_n=0$. Thus every discrete eigenvalue ${\lambda }_n=-{\kappa }_n^2$ is constant.

If ${\phi }_n(x,t)\sim c_n(t)e^{-{\kappa }_{n}x}$ as $x\to +\infty$, the time equation $Q=0$ givesand hence $c_n(t)=c_n(0)e^{4{\kappa }_n^{3}t}$.