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 we obtain
[math]\displaystyle{
\int^\infty_{-\infty}\partial_{t}\lambda w^{2}\,\mathrm{d}x+\left( w\partial_{x}Q-\partial
_{x}wQ\right)\Bigg|^\infty_{-\infty}=0
}[/math] and, since [math]\displaystyle{ w^2\gt 0 }[/math] except for some isolated points,
[math]\displaystyle{
\partial_{t}\lambda=0
}[/math] provided that the eigenfunction [math]\displaystyle{ w }[/math] vanishes at [math]\displaystyle{ x=\pm\infty }[/math] (which is true for the bound
state eigenfunctions, i.e. those from the discrete spectrum). This shows that the discrete eigenvalues are unchanged
and [math]\displaystyle{ u\left( x,t\right) }[/math] evolves according to the KdV. Many other
properties can be found.
Some more details on the content of this lecture can be found in Dunajski, Maciej (2009). Solitons, Instantons, and Twistors (Chapters 2.2 and 2.3).
Scattering Data For the discrete spectrum the eigenfunctions behave like
[math]\displaystyle{
w_{n}\left( x,t\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,t\right) \right) ^{2}\mathrm{d}x=1
}[/math] The continuous spectrum looks like
[math]\displaystyle{
w\left( x,t\right) \approx \mathrm{e}^{-\mathrm{i}kx}+r\left( k,t\right) \mathrm{e}^{\mathrm{i}kx}
,\ \ \ x\rightarrow-\infty
}[/math] [math]\displaystyle{
w\left( x,t\right) \approx a\left( k,t\right) \mathrm{e}^{-\mathrm{i}kx},\ \ \ 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}(t)=k_{n}(0) = 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}+\frac{1}{2\pi}\int_{-\infty}^{\infty}r\left( k,t\right) \mathrm{e}^{\mathrm{i}kx}\mathrm{d}k
}[/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) \mathrm{d}z=0
}[/math] This is a linear integral equation called the 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) }\mathrm{d}z=0
}[/math] From the equation we can see that
[math]\displaystyle{
K\left( x,y,t\right) =-\sum_{n=1}^{N}v_{n}\left( x,t\right) e^{-k_{n}y}
}[/math] If we substitute this into the equation,
[math]\displaystyle{
-\sum_{n=1}^{N}v_{n}\left( x,t\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}v_{m}\left( x,t\right) e^{-k_{m}z}\sum_{n=1}^{N}c_{n}
^{2}\left( t\right) e^{-k_{n}\left( y+z\right) }\mathrm{d}z=0
}[/math] which leads to
[math]\displaystyle{
-\sum_{n=1}^{N} v_{n}\left( x,t\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_{n}^{2}\left(
t\right) }{k_{n}+k_{m}}v_{m}\left( x, t\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] and the
[math]\displaystyle{ e^{-k_{n}y} }[/math] to obtain
[math]\displaystyle{
-v_{n}\left( x,t\right) +c_{n}^2\left( t\right) e^{-k_{n}x}-\sum_{m=1}
^{N}\frac{c_{n}^2\left( t\right) }{k_{n}+k_{m}}
v_{m}\left( x,t\right) e^{-\left( k_{m}+k_{n}\right) x}=0
}[/math] which is an algebraic (finite dimensional system) for the unknowns [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}^2\left( t\right) e^{-k_{m}x} }[/math] and
the elements of [math]\displaystyle{ \mathbf{C} }[/math] are given by
[math]\displaystyle{
c_{mn} = \frac{c_{n}^2\left( t\right)}
{k_{n}+k_{m}}e^{-\left( k_{m}+k_{n}\right) x}
}[/math] This gives us
[math]\displaystyle{
K\left( x,y,t\right) =-\sum_{m=1}^{N} \left(
\mathbf{I}+\mathbf{C}\right) ^{-1}\vec{f}e^{-k_{m}y}
}[/math] We then find [math]\displaystyle{ u(x,t) }[/math] from [math]\displaystyle{ K }[/math] .
Single Soliton Example If [math]\displaystyle{ n=1 }[/math] (a single soliton
solution) we get
[math]\displaystyle{ \begin{matrix}
K\left( x,x,t\right) & =-\frac{c_{1}^2\left( t\right)
e^{-2k_{1}x}}{1+\frac{c_{1}^2\left( t\right) }
{2k_{1}}e^{-2 k_{1} x}}\\
& =\frac{-1}{e^{2k_{1}x - 8k_{1}^{3}t-\alpha} + 1/2k_1}
\end{matrix} }[/math] where [math]\displaystyle{ e^{-\alpha}=1/c_{0}^{2}\left( 0\right) . }[/math] Therefore
[math]\displaystyle{ \begin{align}
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( e^{2k_{1}
x-8k_{1}^{3}t-\alpha} + 1/2k_1\right) ^{2}}\\
& =\frac{8k_{1}^{2}}{\left( \sqrt{2k_{1}}e^{\theta}+e^{-\theta}/\sqrt
{2k_{1}}\right) ^{2}}\\
& =2k_{1}^{2}\mbox{sech}^{2}\left\{ k_{1}\left( x-x_{0}\right) -4k_{1}
^{3}t\right\}
\end{align} }[/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.
Lecture Videos Part 1
Part 2
Part 3
Part 4
