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Use the rapidly decaying real-potential normalization $u_t-6u{u}_x+u_{xxx}=0$. Let $D={\partial }_x$, $L=-D^2+u$ and $A=4D^3-3(uD+Du)=4D^3-6uD-3u_x$. Expanding the commutator cancels its derivative terms and givesThus the KdV equation is the compatibility condition of the Lax pair $L\psi =\lambda \psi$, ${\psi }_t=-A\psi$. The operator $A$ is skew-adjoint on decaying functions. Its evolution transports normalized eigenfunctions unitarily, so the spectrum of $L$ is fixed; alternatively differentiating $L\psi =\lambda \psi$ and using the two equations gives ${\lambda }_t\psi =0$.

At each time solve the KdV Schrodinger spectral problem. For $\lambda =k^2>0$, use the left Jost solution $f_{-}$, normalized as $e^{-ikx}$ at $x\to -\infty$, and write $f_{-}\sim a(k,t)e^{-ikx}+b(k,t)e^{ikx}$ at $+\infty$. The right reflection coefficient is $R=b/a$, and transmission is $1/a$. Because $-A$ tends to $-4D^3$ at infinity, maintaining this Jost normalization gives $f_{-,t}=-A{f}_{-}+4i{k}^3{f}_{-}$. Substitution of its two asymptotic waves yields $a_t=0$ and $b_t=8i{k}^{3}b$. The negative discrete eigenvalues are $-{\kappa }_j^2$, fixed in time. A unit-normalized bound state with right tail $c_j(t)e^{-{\kappa }_{j}x}$ obeys $c_j^{\prime }=4{\kappa }_j^3{c}_j$. Consequently the Time evolution of KdV scattering data isThese signs follow from the specified right-reflection convention; reversing which infinity defines reflection reverses the continuous-data phase convention.

The inverse scattering transform now solves the nonlinear evolution in three steps. Compute $R(k,0)$, the discrete eigenvalues and the norming constants from $u(x,0)$; evolve this KdV scattering data by the simple formulas above; reconstruct $u$ by the Gelfand-Levitan-Marchenko equation. Explicitly setand solve, for $y\ge x$,The recovered potential is $\boxed{u(x,t)=-2dK(x,x,t)/dx}$. Each fixed discrete eigenvalue produces a soliton component, whereas the continuous scattering data encode the dispersive part. The inverse transform, rather than a linear superposition of the potentials, accounts for their nonlinear interaction.

With the sign convention of the displayed operator, the Lax equation is $L_t=[L,A]$. Differentiating $L\phi =k^2\phi$ shows that ${\phi }_t+A\phi$ is another solution at the same spectral parameter. The normalization at $-\infty$ fixessince $A{e}^{-ikx}=4i{k}^3{e}^{-ikx}$ there. At $+\infty$, comparison of the two plane-wave coefficients givesThus the transmission denominator $a$ is time independent. This is the Time evolution of KdV scattering data for the stated normalization.

The logarithmic derivative $S={\phi }_x/\phi +ik$ obeys the Riccati equationIts formal large-$k$ expansion givesThere is an important qualification to the integral identity in the question. For real $k$ with nonzero reflection, $e^{ikx}\phi \sim a+b{e}^{2ikx}$ has no limit at $+\infty$, so the literal improper integral of the exact $S$ need not exist. Continue to $\mathrm{Im}k>0$, where the reflected exponential decays relative to $e^{-ikx}$, and take sufficiently large $k$ away from zeros and with a consistent logarithm. Then the normalization givesThe second equality is an asymptotic expansion, not an asserted convergent series. Time independence of $a$ fixes every coefficient, so each $\int S_{n}dx$ is a conserved quantity of the Korteweg-De Vries equation. The first two nontrivial examples are $-\int udx$ and $\int u^{2}dx$.

For the usual real-valued KdV potential and real formal $k$, write $S=X+iY$ with real coefficient functions. Its imaginary part of the Riccati equation is $Y^{\prime }-2kX+2XY=0$, henceIn the formal expansion, $X$ contains precisely the even-indexed $S_n$ and $Y$ the odd-indexed ones. Rapid decay of $u$ and its derivatives makes every coefficient of the logarithm vanish at both spatial ends. Therefore $\boxed{{\int }_{\mathbb{R}}{S}_{2j}dx=0(j\ge 1)}$. This is a coefficientwise formal identity; it does not claim that the exact real-axis reflected solution has a convergent integral of $X$. The analytically continued scattering identity and the formal expansion are the conventions needed to make the requested argument valid.

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.