Inverse scattering transform

The Inverse Scattering Transform is a procedure for integrating certain nonlinear partial differential equations (pde's) by first converting them into a system of linear ordinary differential equations (ode's). The basic idea is not unlike the Laplace Transform.

Step 1. Determine the nonlinear partial differential equation. The mathematician usually accomplishes this by looking at the physics of the situation.

Step 2. Employ forward scattering. This consists of dreaming up the Lax Pair. The Lax Pair consists of two linear operators, $L$ and $M$, such that $Lv=\lambda v$ and $v_{t}=Mv$. It is extremely important that the eigenvalue $\lambda$ be independent of time; i.e. $\lambda _{t}=0$. In order for this to be the case, take the time derivative of $Lv=\lambda v$ to obtain

$L_{t}v+Lv_{t}=\lambda _{t}v+\lambda v_{t}$.

Plugging in $Mv$ for $v_{t}$ yields

$L_{t}v+LMv=\lambda _{t}v+\lambda Mv$.

Rearranging on the far right term gives us

$L_{t}v+LMv=\lambda _{t}v+MLv$.

Thus,

$L_{t}v+LMv-MLv=\lambda _{t}v.$

Since $v\not =0$, this implies that $\lambda _{t}=0$ if and only if

$L_{t}+LM-ML=0$.

This is Lax's Equation. One important thing to note about Lax's Equation is that $L_{t}$ is the time derivative of $L$ precisely where it explicitly depends on $t$. The reason for defining the differentiation this way is motivated by one very common definition of $L$, which is the Schrödinger operator (see Schrödinger Equation):

$L={\frac {d^{2}}{dx^{2}}}+u.$

Comparing the expression $L_{t}v+Lv_{t}$ with ${\frac {\partial }{\partial t}}({\frac {d^{2}v}{dx^{2}}}+uv)$ shows us that ${\frac {\partial L}{\partial t}}=u_{t}$, thus ignoring the first term.

Once you have concocted the appropriate Lax Pair for your case, it should be that Lax's Equation recovers the original nonlinear pde for you.

Step 3. Determine the time evolution of the eigenvalues $\lambda$, the norming constants, and the reflection coefficient, all three comprising the so-called scattering data. This is all a linear process, though complicated.

Step 4. Perform the inverse scattering procedure by solving the Marchenko Equation, a linear integral equation, to obtain the final solution of the original nonlinear pde. You will need to use all the scattering data to do this. The reader should note that if the reflection coefficient is zero, the process is much easier.

There are many nonlinear pde's of physical interest which the Inverse Scattering Transform solves. A partial list includes the Korteweg - de Vries Equation, the nonlinear Schrödinger equation, coupled nonlinear Schödinger, and the sine - Gordon equation.

References.

M. Ablowitz, H. Segur, Solitons and the Inverse Scattering Transform, SIAM, Philadelphia, 1981.

N. Asano, Y. Kato, Algebraic and Spectral Methods for Nonlinear Wave Equations, Longman Scientific & Technical, Essex, England, 1990.

M. Ablowitz, P. Clarkson, Solitons, Nonlinear Evolution Equations and Inverse Scattering, Cambridge University Press, Cambridge, 1991.

J. Shaw, Mathematical Principles of Optical Fiber Communications, SIAM, Philadelphia, 2004.