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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

We can rewrite (1) as the compatibility condition of two linear operators whose potentials depend non-trivially on the spectral parameter \lambda

[L(\lambda), M(\lambda)] = 0\ \ \ \ (5) ,

where

L = i\partial_x + U\ \ \ \ (6)

M = i\partial_t + V\ \ \ \ (7)

and V will be defined below.

Let \chi(x, t, \lambda) be a fundamental solution of L, i.e. \chi is matrix-valued function whose determinant does not vanish,

L(\lambda)\chi(x, t, \chi)=0, \quad \det \chi(x, t, \chi) \neq 0\ \ \ \ (8).

From the compatibility condition (5) we get

\left[L(\lambda), M(\lambda)\right]\chi(x, t, \lambda)=\\=L(\lambda)M(\lambda)\chi(x,t,\lambda)-M(\lambda)L(\lambda)\chi(x, t, \lambda) \\= L(\lambda)M(\lambda)\chi(x, t, \lambda)=0\ \ \ \ (9)

which says that if \chi(x, t, \lambda) is a fundamental solution of
L(\lambda) then M(\lambda)\chi(x, t, \lambda) is also a
fundamental solution of L(\lambda).

Proposition. Equation (5) is equivalent to the generalization of the NLS equation if U and V are of the form
U = q - \lambda \sigma_3\ \ \ \ (10) \\ V = -i \sigma_3 q_x - q^+ q^- \sigma_3 - 2\lambda q + 2 \lambda^2 \sigma_3\ \ \ \ (11)

Proof

Start with substituting (6) and (7) into (5).

\left[i \partial_x + U, i \partial_t + V\right] \Psi = 0 \\ \nonumber (i \partial_x + U)(i \partial_t + V)\Psi - (i \partial_t + V)(i \partial_x + U) \Psi = 0 \\ \nonumber (i \partial_x + U)(i \Psi_t + V\Psi) - (i \partial_t + V)(i \Psi_x + U\Psi) = 0 \nonumber

From there, since \Psi is an arbitrary function we get

iV_{x} - iU_{t} +\left[U, V\right]=0\ \ \ \ (12).

We can rewrite (10) and (11) in the matrix form

U = q-\lambda \sigma_3 = \begin{bmatrix} 0 & q^{+} \\ q^{-} & 0\end{bmatrix} - \begin{bmatrix} \lambda & 0 \\ 0 & -\lambda \end{bmatrix} = \begin{bmatrix} -\lambda & q^{+} \\ q^{-} & \lambda \end{bmatrix}\ \ \ \ (13)

V = \begin{bmatrix} -q^+q^- + 2\lambda^2 & -iq^{+}_x - 2\lambda q^+ \\ iq^{-}_x - 2\lambda q^- & q^+q^- - 2\lambda^2 \end{bmatrix}\ \ \ \ (14)

Inserting (13) and (14) into (12) leads us to

iq_t^{+} + q_{xx}^{+} + 2q^{+} q^{-} q^{+}(x, t) = 0

-iq_t^{-} + q_{xx}^{-} + 2q^{-} q^{+} q^{-}(x, t) = 0

Then taking reduction

q^{+} = \pm (q^{-})^{*} = q

we get

-iq_t + q_{xx} \pm 2|q|^2 q(x, t) = 0 \ \ \ \ (15)

which is generalization of the NLS equation. Equation with + sign next to nonlinear factor is called defocusing and the one with the - sign is called focusing.

From this we can see that the Lax representation (5) with the choices (6) and (7) for L and M in is equivalent to the system (15).

QED

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].