Difference between revisions of "Introduction to the Inverse Scattering Transform"
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The Miura transformation is given by | The Miura transformation is given by | ||
<center><math> | <center><math> | ||
− | u=-v^{2}- | + | u=-v^{2}-\partial_x v\, |
</math></center> | </math></center> | ||
and if <math>v</math> satisfies the mKdV | and if <math>v</math> satisfies the mKdV |
Revision as of 06:38, 9 September 2020
Nonlinear PDE's Course | |
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Current Topic | Introduction to the Inverse Scattering Transform |
Next Topic | Properties of the Linear Schrodinger Equation |
Previous Topic | 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
with [math]\displaystyle{ \left\vert u\right\vert \rightarrow0 }[/math] as [math]\displaystyle{ x\rightarrow\pm\infty. }[/math]
The Miura transformation is given by
and if [math]\displaystyle{ v }[/math] satisfies the mKdV
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 known transformation which linearises this equation. It we write
then we obtain the equation
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
The eigenfunctions and eigenvalues of this scattering problem play a key role in the inverse scattering transformation. Note that this is Schrodinger's equation.