The Dressing Method

30 Friday Dec 2016

This is the third part in a series of Lax Pair and Nonlinear Schrödinger (NLS) Equation (view the previous post here )

In this section we will study the ZS system where the Lax operator has the form:

$L\chi \equiv (i\frac{d}{dx} + q(x) - \lambda \sigma_3)\chi(x, \lambda) = 0\ \ \ \ (16)$

where

$\lim \limits_{x \to \pm \infty} q(x) = q_{\pm} = \begin{bmatrix} 0 & q_{\pm} \\ -q_{\pm}^{*} & 0 \\ \end{bmatrix}$

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

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

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