= Solution
With the sign convention of the displayed operator, the <Lax equation> is $L_t=[L,A]$. Differentiating $L\varphi=k^2\varphi$ shows that $\varphi_t+A\varphi$ is another solution at the same spectral parameter. The normalization at $-\infty$ fixes
$$
\varphi_t=-A\varphi+4ik^3\varphi,
$$
since $A e^{-ikx}=4ik^3e^{-ikx}$ there. At $+\infty$, comparison of the two plane-<wave> coefficients gives
$$
\boxed{a_t=0,\qquad b_t=8ik^3b,\qquad b(k,t)=b(k,0)e^{8ik^3t}.}
$$
Thus the transmission denominator $a$ is time independent. This is the <Time evolution of KdV scattering data> for the stated normalization.
The <logarithmic derivative> $S=\varphi_x/\varphi+ik$ obeys the <Riccati equation>
$$
S'-2ikS+S^2=u.
$$
Its formal large-$k$ expansion gives
$$
S_1=-u,\qquad S_{n+1}=S_n'+\sum_{j=1}^{n-1}S_jS_{n-j},\qquad
S_2=-u',\quad S_3=-u''+u^2,\quad S_4=-u'''+4uu'.
$$
There is an important qualification to the <integral> identity in the question. For real $k$ with nonzero reflection, $e^{ikx}\varphi\sim a+b e^{2ikx}$ has no <limit> at $+\infty$, so the literal improper <integral> of the exact $S$ need not exist. Continue to $\operatorname{Im}k>0$, 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 gives
$$
\boxed{a(k,t)=\exp\left(\int_{-\infty}^{\infty}S(x,k,t)\,dx\right),\qquad
\log a\sim\sum_{n\geq1}\frac{\int S_n\,dx}{(2ik)^n}.}
$$
The 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 $-\int u dx$ and $\int u^2dx$.
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 $Y'-2kX+2XY=0$, hence
$$
X=\frac{Y'}{2(k-Y)}=-\frac12\partial_x\log(1-Y/k).
$$
In 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 $\boxed{\int_{\mathbb R}S_{2j}\,dx=0\quad(j\geq1)}$. 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.
Back to article page
