Introduction to the Inverse Scattering Transform
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]\left\vert u\right\vert \rightarrow0[/math] as [math]x\rightarrow\pm\infty.[/math]
The Miura transformation is given by
and if [math]v[/math] satisfies the mKdV
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
then we obtain the equation
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
The eigenfunctions and eigenvalues of this scattering problem play a key role in the inverse scattering transformation. Note that this is Schrodinger's equation.