Introduction to the Inverse Scattering Transform

Nonlinear PDE's Course
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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

[math]\displaystyle{ \begin{matrix} \partial_{t}u+6u\partial_{x}u+\partial_{x}^{3}u & =0\\ u(x,0) & =f\left( x\right) \end{matrix} }[/math]

with [math]\displaystyle{ \left\vert u\right\vert \rightarrow0 }[/math] as [math]\displaystyle{ x\rightarrow\pm\infty. }[/math]

The Miura transformation is given by

[math]\displaystyle{ u=v^{2}+v_{x} }[/math]

and if [math]\displaystyle{ v }[/math] satisfies the mKdV

[math]\displaystyle{ \partial_{t}v-6v^{2}\partial_{x}v+\partial_{x}^{3}v=0 }[/math]

then [math]\displaystyle{ u }[/math] satisfies the KdV (but not vice versa). We can think about the Miura transformation as being a nonlinear ODE solving for [math]\displaystyle{ v }[/math] given [math]\displaystyle{ 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

[math]\displaystyle{ v=\frac{\left( \partial_{x}w\right) }{w} }[/math]

then we obtain the equation

[math]\displaystyle{ \partial_{x}^{2}w+uw=0 }[/math]

The KdV is invariant under the transformation [math]\displaystyle{ x\rightarrow x+6\lambda t, }[/math] [math]\displaystyle{ u\rightarrow u+\lambda. }[/math] Therefore we consider the associated eigenvalue problem

[math]\displaystyle{ \partial_{x}^{2}w+uw=-\lambda w }[/math]

The eigenfunctions and eigenvalues of this scattering problem play a key role in the inverse scattering transformation. Note that this is Schrodinger's equation.

Properties of the eigenfunctions

The equation

[math]\displaystyle{ \partial_{x}^{2}w+uw=-\lambda w }[/math]

has two kinds of solutions for [math]\displaystyle{ u\rightarrow0 }[/math] as [math]\displaystyle{ x\rightarrow\pm\infty. }[/math] The first are waves and the second are bound solutions. It is well known that there are at most a finite number of bound solutions (provided [math]\displaystyle{ u\rightarrow0 }[/math] as [math]\displaystyle{ x\pm\infty }[/math] sufficiently rapidly) and a continum of solutions for the incident waves.

Example: Scattering by a Well

The properties of the eigenfunction is prehaps seem most easily through the following example

[math]\displaystyle{ u\left( x\right) =\left\{ \begin{matrix} 0 & x\notin\left[ -1,1\right] \\ b & x\in\left[ -1,1\right] \end{matrix} \right. }[/math]

where [math]\displaystyle{ b\gt 0. }[/math]

\paragraph{Case when [math]\displaystyle{ \lambda\gt 0 }[/math]}

If we solve this equation for the case when [math]\displaystyle{ \lambda\lt 0, }[/math] [math]\displaystyle{ \lambda=-k^{2} }[/math] we get

[math]\displaystyle{ w\left( x\right) =\left\{ \begin{matrix} [c] a_{1}e^{kx}, & x\lt -1\\ b_{1}\cos\kappa x+b_{2}\sin\kappa x & -1\lt x\lt 1\\ a_{2}e^{-kx} & x\gt 1 \end{matrix} \right. }[/math]

where [math]\displaystyle{ \kappa=\sqrt{b-k^{2}} }[/math] where we have assumed that [math]\displaystyle{ b\gt k^{2} }[/math] (there is no solution for [math]\displaystyle{ b\lt k^{2}). }[/math] We then match [math]\displaystyle{ w }[/math] and its derivative at [math]\displaystyle{ x=\pm1 }[/math] to solve for [math]\displaystyle{ a }[/math] and [math]\displaystyle{ b }[/math]. This leads to two system of equation, one for the even ([math]\displaystyle{ a_{1}=a_{2} }[/math] and [math]\displaystyle{ b_{2}=0 }[/math] ) and one for the odd solutions ([math]\displaystyle{ a_{1}=-a_{2} }[/math] and [math]\displaystyle{ b_{1}=0) }[/math]. The solution for the even solutions is

[math]\displaystyle{ \left( \begin{matrix} [c] e^{-kx} & -\cos\kappa\\ ke^{-kx} & \sin\kappa \end{matrix} \right) \left( \begin{matrix} [c] a_{1}\\ b_{1} \end{matrix} \right) =\left( \begin{matrix} [c] 0\\ 0 \end{matrix} \right) }[/math]

This can non trivial solutions when

[math]\displaystyle{ \det\left( \begin{matrix} [c] e^{-kx} & -\cos\kappa\\ ke^{-kx} & \sin\kappa \end{matrix} \right) =0 }[/math]

which gives us the equation

[math]\displaystyle{ e^{-kx}\sin\kappa+\left( \cos\kappa\right) ke^{-kx}=0 }[/math]

or

[math]\displaystyle{ \tan\kappa=-k=-\sqrt{b-\kappa^{2}} }[/math]

We know that [math]\displaystyle{ 0\lt \kappa\lt \sqrt{b} }[/math] and if we plot this we see that we obtain a finite number of solutions.

\paragraph{Case when [math]\displaystyle{ \lambda\gt 0 }[/math]}

When [math]\displaystyle{ \lambda\gt 0 }[/math] we write [math]\displaystyle{ \lambda=k^{2} }[/math] and we obtain solution

[math]\displaystyle{ w\left( x\right) =\left\{ \begin{matrix} [c] e^{-ikx}+re^{ikx}, & x\lt -1\\ b_{1}\cos\kappa x+b_{2}\sin\kappa x & -1\lt x\lt 1\\ ae^{-ikx} & x\gt 1 \end{matrix} \right. }[/math]

where [math]\displaystyle{ \kappa=\sqrt{b+k^{2}}. }[/math] Matching [math]\displaystyle{ w }[/math] and its derivaties at [math]\displaystyle{ x=\pm1 }[/math] we obtain

[math]\displaystyle{ \left( \begin{matrix} [c] -e^{-ik} & \cos\kappa & -\sin\kappa & 0\\ ike^{-ik} & \kappa\sin\kappa & \kappa\cos\kappa & 0\\ 0 & \cos\kappa & \sin\kappa & -e^{-ik}\\ 0 & -\kappa\sin\kappa & \kappa\cos\kappa & ike^{-ik} \end{matrix} \right) \left( \begin{matrix} [c] r\\ b_{1}\\ b_{2}\\ a \end{matrix} \right) =\left( \begin{matrix} [c] e^{ik}\\ ike^{-ik}\\ 0\\ 0 \end{matrix} \right) }[/math]

Connection with the KdV

If we substitute the relationship

[math]\displaystyle{ \partial_{x}^{2}w+uw=-\lambda w }[/math]

into the KdV after some manipulation we obtain

[math]\displaystyle{ \partial_{t}\lambda w^{2}+\partial_{x}\left( w\partial_{x}Q-\partial _{x}wQ\right) =0 }[/math]

where [math]\displaystyle{ Q=\partial_{t}w+\partial_{x}^{3}w-3\left( \lambda-u\right) \partial_{x}w. }[/math] If we integrate this equation then we obtain the result that

[math]\displaystyle{ \partial_{t}\lambda=0 }[/math]

provided that the eigenfunction [math]\displaystyle{ w }[/math] is bounded (which is true for the bound state eigenfunctions). This shows that the discrete eigenvalues are unchanged and [math]\displaystyle{ u\left( x,t\right) }[/math] evolves according to the KdV.

Scattering Data

For the discrete spectrum the eigenfunctions behave like

[math]\displaystyle{ w_{n}\left( x\right) =c_{n}\left( t\right) e^{-k_{n}x} }[/math]

as [math]\displaystyle{ x\rightarrow\infty }[/math] with

[math]\displaystyle{ \int_{-\infty}^{\infty}\left( w_{n}\left( x\right) \right) ^{2}dx=1 }[/math]

The continuous spectrum looks like

[math]\displaystyle{ v\left( x,t\right) \approx e^{-ikx}+r\left( k,t\right) e^{ikx} ,\ \ \ x\rightarrow-\infty }[/math] [math]\displaystyle{ v\left( x,t\right) \approx a\left( k,t\right) e^{-ikx},\ \ \ x\rightarrow \infty }[/math]

where [math]\displaystyle{ r }[/math] is the reflection coefficient and [math]\displaystyle{ a }[/math] is the transmission coefficient. This gives us the scattering data at [math]\displaystyle{ t=0 }[/math]

[math]\displaystyle{ S\left( \lambda,0\right) =\left( \left\{ k_{n},c_{n}\left( 0\right) \right\} _{n=1}^{N},r\left( k,0\right) ,a\left( k,0\right) \right) }[/math]

The scattering data evolves as

[math]\displaystyle{ k_{n}=k_{n} }[/math] [math]\displaystyle{ c_{n}\left( t\right) =c_{n}\left( 0\right) e^{4k_{n}^{3}t} }[/math] [math]\displaystyle{ r\left( k,t\right) =r\left( k,0\right) e^{8ik^{3}t} }[/math] [math]\displaystyle{ a\left( k,t\right) =a\left( k,0\right) }[/math]

We can recover [math]\displaystyle{ u }[/math] from scattering data. We write

[math]\displaystyle{ F\left( x,t\right) =\sum_{n=1}^{N}c_{n}^{2}\left( t\right) e^{-k_{n} x}+\int_{-\infty}^{\infty}r\left( k,t\right) e^{ikx}dk }[/math]

Then solve

[math]\displaystyle{ K\left( x,y;t\right) +F\left( x+y;t\right) +\int_{x}^{\infty}K\left( x,z;t\right) F\left( z+y;t\right) dz=0 }[/math]

This is a linear integral equation called the \emph{Gelfand-Levitan-Marchenko }equation. We then find [math]\displaystyle{ u }[/math] from

[math]\displaystyle{ u\left( x,t\right) =2\partial_{x}K\left( x,x,t\right) }[/math]

Reflectionless Potential

In general the IST is difficult to solve. However, there is a simplification we can make when we have a reflectionless potential (which we will see gives rise to the soliton solutions). The reflectionless potential is the case when [math]\displaystyle{ r\left( k,0\right) =0 }[/math] for all values of [math]\displaystyle{ k }[/math] for some [math]\displaystyle{ u. }[/math] In this case

[math]\displaystyle{ F\left( x,t\right) =\sum_{n=1}^{N}c_{n}^{2}\left( t\right) e^{-k_{n}x} }[/math]

then

[math]\displaystyle{ K\left( x,y,t\right) +\sum_{n=1}^{N}c_{n}^{2}\left( t\right) e^{-k_{n}\left( x+y\right) }+\int_{x}^{\infty}K\left( x,z,t\right) \sum_{n=1}^{N}c_{n}^{2}\left( t\right) e^{-k_{n}\left( y+z\right) }dz=0 }[/math]

From the equation we can see that

[math]\displaystyle{ K\left( x,y,t\right) =-\sum_{m=1}^{N}c_{m}\left( t\right) v_{m}\left( x\right) e^{-k_{m}y} }[/math]

If we substitute this into the equation

[math]\displaystyle{ -\sum_{n=1}^{N}c_{n}\left( t\right) v_{n}\left( x\right) e^{-k_{n}y} +\sum_{n=1}^{N}c_{n}^{2}\left( t\right) e^{-k_{n}\left( x+y\right) } +\int_{x}^{\infty}-\sum_{m=1}^{N}c_{m}\left( t\right) v_{m}\left( x\right) e^{-k_{m}y}\sum_{n=1}^{N}c_{n}^{2}\left( t\right) e^{-k_{n}\left( y+z\right) }dz=0 }[/math]

which leads to

[math]\displaystyle{ -\sum_{n=1}^{N}c_{n}\left( t\right) v_{n}\left( x\right) e^{-k_{n}y} +\sum_{n=1}^{N}c_{n}^{2}\left( t\right) e^{-k_{n}\left( x+y\right) } -\sum_{n=1}^{N}\sum_{m=1}^{N}\frac{c_{m}\left( t\right) c_{n}^{2}\left( t\right) }{k_{n}+k_{m}}v_{m}\left( x\right) e^{-k_{m}x}e^{-k_{n}\left( y+x\right) }=0 }[/math]

and we can eliminate the sum over [math]\displaystyle{ n }[/math] , the [math]\displaystyle{ c_{n}\left( t\right) , }[/math] and the [math]\displaystyle{ e^{-k_{n}y} }[/math] to obtain

[math]\displaystyle{ -v_{n}\left( x\right) +c_{n}\left( t\right) e^{-k_{n}x}-\sum_{m=1} ^{N}\frac{c_{n}\left( t\right) c_{m}\left( t\right) }{k_{n}+k_{m}} v_{m}\left( x\right) e^{-\left( k_{m}+k_{n}\right) x}=0 }[/math]

which is an algebraic (finite dimensional system)\ for the unknows [math]\displaystyle{ v_{n}. }[/math] We can write this as

[math]\displaystyle{ \left( \mathbf{I}+\mathbf{C}\right) \vec{v}=\vec{f} }[/math]

where [math]\displaystyle{ f_{m}=c_{m}\left( t\right) e^{-k_{m}x} }[/math] and

[math]\displaystyle{ c_{mn}=\sum_{m=1}^{N}\frac{c_{n}\left( t\right) c_{m}\left( t\right) }{k_{n}+k_{m}}e^{-\left( k_{m}+k_{n}\right) x} }[/math] [math]\displaystyle{ K\left( x,y,t\right) =-\sum_{m=1}^{N}c_{m}\left( t\right) \left( \mathbf{I}+\mathbf{C}\right) ^{-1}\vec{f}e^{-k_{m}y} }[/math]

This leads to

[math]\displaystyle{ u\left( x,t\right) =2\partial_{x}^{2}\log\left[ \det\left( \mathbf{I} +\mathbf{C}\right) \right] }[/math]

Lets consider some simple examples. First of all if [math]\displaystyle{ n=1 }[/math] (the single soliton solution) we get

[math]\displaystyle{ \begin{matrix} K\left( x,x,t\right) & =-\frac{c_{1}\left( t\right) c_{1}\left( t\right) e^{-k_{1}x}e^{-k_{1}x}}{1+\frac{c_{1}\left( t\right) c_{1}\left( t\right) }{k_{1}+k_{1}}e^{-\left( k_{1}+k_{1}\right) x}}\\ & =\frac{-1}{1+e^{2k_{1}x-8k_{1}^{3}t-\alpha}} \end{matrix} }[/math]

where [math]\displaystyle{ e^{-\alpha}=2c_{0}^{2}\left( 0\right) . }[/math] Therefore

[math]\displaystyle{ \begin{matrix} u\left( x,t\right) & =2\partial_{x}K\left( x,x,t\right) \\ & =\frac{4k_{1}e^{2k_{1}x-8k_{1}^{3}t-\alpha}}{\left( 1+e^{2k_{1} x-8k_{1}^{3}t-\alpha}\right) ^{2}}\\ & =\frac{-8k_{1}^{2}}{\left( \sqrt{2k_{1}}e^{\theta}+e^{-\theta} /\sqrt{2k_{1}}\right) ^{2}}\\ & =2k^{2}\sec^{2}\left\{ k_{1}\left( x-x_{0}\right) -4k_{1}^{3}t\right\} \end{matrix} }[/math]

where [math]\displaystyle{ \theta=k_{1}x-4k^{3}t-\alpha/2 }[/math] and [math]\displaystyle{ \sqrt{2k}e^{-\alpha/2}=e^{-kx_{0} } }[/math]. This is of course the single soliton solution.