Question for inverse scattering method I'm now reading The Korteweg-de Vries Equation: a survey of results,written by Robert M.Miura. When I read the part of inverse scattering method. It says: for the continuous spectrum, we can assume the eigenvalue $\lambda=k^2$ is a constant.I don't understand why he can make such assumption.Thanks.
 A: The scattering data of the KdV are 


*

*the discrete eigenvalues $\lambda_{n} = -(k^{2}_{n}) ,$  where $n = 1,2,\ldots N$, and

*the functions $a(k;t), b(k;t)$ - also known as the reflection and transmission coefficients - that are defined for non-negative $k$.


We are interested in the time-evolution of this data. 
With the discrete (i.e. negative) eigenvalues we're interested in their actual values. We can do a little bit of legwork (which I assume Miura covers, if not check Drazin and Johnson's 'Solitons: an introduction') to show that these are constant.
For the continuous (i.e. positive) spectrum we don't care about the eigenvalues, we care about the values of the reflection and transmission coefficients at a particular value of $k$. We don't care about the eigenvalue because we know a priori that any positive $k$ has a continuous eigenfunction - the interesting thing is how that eigenfunction behaves; how it scatters. Thus we fix $k$ (assume that it's constant) and investigate $a(k;t), b(k;t)$. Again, a little bit of work shows us that $a(k;t) = a(k;0)$ and $b(k;t) = b(k;0)\exp(8ik^{3}t).$
