View source for Introduction to the Inverse Scattering Transform

{{nonlinear waves course
 | chapter title = Introduction to the Inverse Scattering Transform
 | next chapter = [[Properties of the Linear Schrodinger Equation]]
 | previous chapter = [[Conservation Laws for the KdV]]
}}


The inverse scattering transformation gives a way to solve the KdV equation
exactly. You can think about is as being an analogous transformation to the
Fourier transformation, except it works for a non linear equation. We want to
be able to solve
<center><math>\begin{matrix}
\partial_{t}u+6u\partial_{x}u+\partial_{x}^{3}u  &  =0\\
u(x,0)  &  =f\left(  x\right)
\end{matrix}</math></center>
with <math>\left\vert u\right\vert \rightarrow0</math> as <math>x\rightarrow\pm\infty.</math>

The Miura transformation is given by
<center><math>
u=-v^{2}-v_{x}\,
</math></center>
and if <math>v</math> satisfies the mKdV
<center><math>
\partial_{t}v-6v^{2}\partial_{x}v+\partial_{x}^{3}v=0
</math></center>
then <math>u</math> satisfies the KdV (but not vice versa). We can think about the Miura
transformation as being a nonlinear ODE solving for <math>v</math> given <math>u.</math> This
nonlinear ODE is also known as the Riccati equation and there is a well know
transformation which linearises this equation. It we write
<center><math>
v=\frac{\left(  \partial_{x}w\right)  }{w}
</math></center>
then we obtain the equation
<center><math>
\partial_{x}^{2}w+uw=0
</math></center>
The KdV is invariant under the transformation <math>x\rightarrow x+6\lambda t,</math>
<math>u\rightarrow u+\lambda.</math> Therefore we consider the associated eigenvalue
problem
<center><math>
\partial_{x}^{2}w+uw=-\lambda w
</math></center>
The eigenfunctions and eigenvalues of this scattering problem play a key role
in the inverse scattering transformation. Note that this is Schrodinger's equation.