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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.

Consider the Lax Representation

iL_t = [L, M] \ \ \ \ (1)

where

[L, M] := LM - ML

is the commutator and L and M are time dependent linear operators acting on a Hilbert space. Zakharov and Shabat have chosen L-operator as

L\Psi \equiv (i\frac{d}{dx} + U(x,t,\lambda))\Psi(x, t, \lambda) = 0\ \ \ \ (2)

where \Psi is an eigenfunction and \lambda is corresponding eigenvalue. Additionally,

U(x,y,\lambda) = q(x,t) - \lambda \sigma_3\ \ \ \ (3)

q(x, t) = \left[\begin{array}{cc} 0 & q^{+} \\ q^{-} & 0 \\ \end{array}\right]\ \ \ \ (4)

where

q^{+}(x,t) = u(x, t) 

and

q^{-}(x,t) = u^{*}(x, t) 

Here q^{+} and q^{-} are mutually complex conjugated pairs, q plays the role of a potential and

\sigma_3 = \left[\begin{array}{cc} 1 & 0 \\ 0 & -1 \\ \end{array}\right]

is a Pauli matrix. Pauli matrices are a set of three 2 \times 2 complex matrices which are Hermitian and unitary. We denote them with \sigma_1, \sigma_2 and \sigma_3, where \sigma_3 is the same as above and

\sigma_1 = \begin{bmatrix} 0 & 1 \\ 1 & 0 \\ \end{bmatrix}, \quad \sigma_2 = \begin{bmatrix} 0 & -i \\ i & 0 \\ \end{bmatrix}.

Pauli matrices have the property

[\sigma_j, \sigma_k]=2i\epsilon_{j k l}\sigma_l, \quad j, k, l = 1, 2, 3.

Also

q_{0}(x, t) = q^{+}\sigma_{+} + q^{-}\sigma_{-}

where

\sigma_+ = \begin{bmatrix} 0 & 1 \\ 0 & 0 \\ \end{bmatrix}, \quad \sigma_{-} = \begin{bmatrix} 0 & 0 \\ 1 & 0 \\ \end{bmatrix}.

The system of equations (2) - (4) is called the Zakharov-Shabat (ZS) system.

Resources:

  1. V. S. Gerdjikov, G. Vilasi, A. B. Yanovski, Integrable Hamiltonian Hierarchies. Spectral and Geometric Methods, Springer, (2008), pp. 3-104.
  2. Mark J. Ablowitz, Nonlinear Dispersive Waves: Asymptotic Analysis and Solitons, Cambridge, (2011), pp. 1-22, 54-61, 81-86, 169-182.
  3. D. J. Korteweg, G. De Vries, On the change of form of long waves advancing in a rectangular canal, and on a new type of long stationary waves, Philosophical Magazine Series 5, Volume 39, Issue 240, (1895).
  4. Y.Ben-Aryeh, Soliton solutions of nonlinear Schrödinger (NLS) and Korteweg de Vries (KdV) equations related to zero curvature in the x,t plane. Arxiv: 1111.5226 [pdf].