Past exam of the mathematics course of the University of Cambridge / 2020 / ii / Paper 2 / 33C (Integrable Systems)

The Korteweg-De Vries equation is the compatibility condition $L_t=[P,L]$ for the Lax pairFor rapidly decaying initial data, first solve the direct KdV Schrodinger spectral problemComparing its Jost solutions at the two spatial infinities produces the KdV scattering data: the reflection coefficient $R(k)$ on the continuous spectrum, discrete eigenvalues $-{\kappa }_n^2$, and norming amplitudes $c_n$ for the bound states.

The Lax evolution is isospectral. More precisely, the Time evolution of KdV scattering data isThis is the simple linear evolution that makes the nonlinear initial-value problem tractable.

To invert the data, formSolve the Gelfand-Levitan-Marchenko equationand reconstructDirect scattering at $t=0$, linear evolution of the data, and this inverse step constitute the inverse scattering transform.

For one bound state at $-{\kappa }^2$ and no reflection, writeThe rank-one Gelfand-Levitan-Marchenko kernel has the form $K(x,y;t)=-A(x,t)e^{-\kappa y}$. Substitution into the integral equation givesand henceChoose $x_0$ by $C_0/(2\kappa )=e^{2\kappa x_0}$. ThenDifferentiating the reconstruction formula gives the one-soliton solutionIt is a negative soliton of amplitude $2{\kappa }^2$ travelling to the right with speed $4{\kappa }^2$, as required by this sign convention for KdV.

The first Born approximation for a one-dimensional reflection coefficient follows by replacing the Jost solution inside its Volterra scattering equation by the free wave $e^{ikx}$:

To check the inverse problem, any shallow discrete state contributes only at $O({ϵ}^2)$ pointwise, so to first order the Marchenko input isDifferentiating and inserting the Born approximation giveswhere the second line is the Fourier inversion theorem. At first order the quadratic integral term in the Marchenko equation may be dropped, soThe reconstruction formula therefore returnsThus direct and inverse scattering agree to first order.