The <Korteweg-De Vries equation> is the compatibility condition $L_t=[P,L]$ for the <Lax pair>
$$
L=-\partial_x^2+u,
\qquad
P=-4\partial_x^3+6u\partial_x+3u_x.
$$
For rapidly decaying initial data, first solve the direct <KdV Schrodinger spectral problem>
$$
L\psi=k^2\psi.
$$
Comparing its <Jost solution>[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> is
$$
\kappa_n(t)=\kappa_n(0),\qquad
R(k,t)=R(k,0)e^{8ik^3t},\qquad
c_n(t)=c_n(0)e^{4\kappa_n^3t}.
$$
This is the simple linear evolution that makes the nonlinear initial-value problem tractable.

To invert the data, form
$$
F(s,t)
=\frac1{2\pi}\int_{-\infty}^{\infty}
R(k,t)e^{iks}\,dk
+\sum_n c_n(t)^2e^{-\kappa_ns}.
$$
Solve the <Gelfand-Levitan-Marchenko equation>
$$
K(x,y;t)+F(x+y,t)
+\int_x^\infty K(x,z;t)F(z+y,t)\,dz=0,
\qquad y\geq x,
$$
and reconstruct
$$
\boxed{u(x,t)=-2\frac{\partial}{\partial x}K(x,x;t)}.
$$
Direct scattering at $t=0$, linear evolution of the data, and this inverse step constitute the <inverse scattering transform>.

Solution

Past exam of the mathematics course of the University of Cambridge / 2020 / ii / Paper 2 / 33C / i / Solution

{wiki}

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.

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 Incoming links: Jost solution