Lax Pair and The Nonlinear Schrödinger (NLS) Equation (Part 1)

30 Friday Dec 2016

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The idea behind Lax Representation (Lax Pair) is that the nonlinear PDE can be expressed as a compatibility condition of two linear equations.

If we have a system of two partial differential equations

$\begin{array}{ccc} \left( i \partial_x + U(x, \lambda) \right) \Psi & = &0 \\ \left( i \partial_t + V(x, \lambda) \right) \Psi & = & 0 \end{array}$

on a domain $\Omega \subset \mathbb{R}^2$ , we can define their compatibility condition as

$i V_x - iU_t + [U, V] = 0.$

Lax Pair and the Nonlinear Schrödinger (NLS) Equation (Part 2)

30 Friday Dec 2016

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Operator L has continuous and discrete spectrum in general so we can separate solutions into two groups:

Solutions parametrized by data on the continuous spectrum only
Solutions parametrized by the data on the discrete spectrum, which are known as soliton solutions