Inverse Scattering Transform - Method of Solution

Method of Solution

Step 1. Determine the nonlinear partial differential equation. This is usually accomplished by analyzing the physics of the situation being studied.

Step 2. Employ forward scattering. This consists in finding the Lax pair. The Lax pair consists of two linear operators, and, such that and . It is extremely important that the eigenvalue be independent of time; i.e. Necessary and sufficient conditions for this to occur are determined as follows: take the time derivative of to obtain

Plugging in for yields

Rearranging on the far right term gives us

Thus,

Since, this implies that if and only if

This is Lax's equation. One important thing to note about Lax's equation is that is the time derivative of precisely where it explicitly depends on . The reason for defining the differentiation this way is motivated by the simplest instance of, which is the Schrödinger operator (see Schrödinger equation):

where u is the "potential". Comparing the expression with shows us that thus ignoring the first term.

After concocting the appropriate Lax pair it should be the case that Lax's equation recovers the original nonlinear PDE.

Step 3. Determine the time evolution of the eigenfunctions associated to each eigenvalue, the norming constants, and the reflection coefficient, all three comprising the so-called scattering data. This time evolution is given by a system of linear ordinary differential equations which can be solved.

Step 4. Perform the inverse scattering procedure by solving the Gelfand–Levitan–Marchenko integral equation (Israel Moiseevich Gelfand and Boris Moiseevich Levitan; Vladimir Aleksandrovich Marchenko), a linear integral equation, to obtain the final solution of the original nonlinear PDE. All the scattering data is required in order to do this. Note that if the reflection coefficient is zero, the process becomes much easier. Note also that this step works if is a differential or difference operator of order two, but not necessarily for higher orders. In all cases however, the inverse scattering problem is reducible to a Riemann–Hilbert factorization problem. (See Ablowitz-Clarkson (1991) for either approach. See Marchenko (1986) for a mathematical rigorous treatment.)