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}

We assume that potential q(x) satisfies the following condition:

  • q(x) belongs to the space $latex $\mathcal{M}$ of off-diagonal 2 \times 2 matrix valued complex infinitely differentiable functions tending to 0 for |x| \rightarrow \infty faster than any negative power of x. This type of functions are called Schwartz-type functions. These properties are satisfied for all values of t.

We will define scattering operator L as

L(\lambda) = i\frac{d}{dx} + q(x) - \lambda \sigma_3.

The spectrum of L(\lambda) consists of a continuous and a discrete part and the equation (16) is equivalent to an eigenvalue problem

L\chi = i\left(\frac{d}{dx} + q(x) - \lambda \sigma_{3} \right)\chi(x, \lambda) = 0

and therefore its spectrum must be on the real axis. The continuous part of its
spectrum is determined by equality Im\lambda = 0 (\cite{ivanov}, \cite{valchev}).
In the Inverse Scattering Theory there are special fundamental solutions \phi and \psi, called Jost solutions, defined by their asymptotics as follows:

\lim_{x \to \infty} e^{i \lambda \sigma_3 x} \Psi (x, t \lambda) = 1\!\!1\ \ \ \ (17)

\lim_{x \to -\infty} e^{i \lambda \sigma_3 x} \Phi (x, t \lambda) = 1\!\!1\ \ \ \ (18).

Both solutions have determinants equal to 1 so they are fundamental solutions and linearly related. This means that there exists the transition matrix between the Jost solutions, called scattering matrix T(t, \lambda). The Jost solutions and the scattering matrix are defined for \lambda on the continuous spectrum only. The details can be found in \cite{vladimir}.
The scattering matrix is defined by:

T(t, \lambda) = \begin{bmatrix} a^+(\lambda) & -b^-(t, \lambda) \\ b^+(t, \lambda) & a^-(\lambda) \end{bmatrix}, \quad \lambda \in \mathbb{R}\ \ \ \ (19)

and Jost solutions are related to the scattering matrix by

\Phi(x, \lambda) = \Psi(x, \lambda) T(\lambda), \quad \lambda \in \mathbb{R}\ \ \ \ (20).

There also exist a fundamental analytic solutions (FAS) (eigenfunctions) of the ZS system and we denote them by \chi^ \pm (x, \lambda). They are analytic functions of \lambda for $\chi^ + (x, \lambda)$, which is defined on \mathbb{C}^+, where Im \lambda \geq 0. Similarly, \chi^ - (x, \lambda) is defined on \mathbb{C}^-, where Im \lambda \leq 0. FAS satisfy the Riemann Hilbert Problem (RHP) defined as

\chi^+(x) = \chi^-(\lambda, x)G(\lambda), \quad Im \lambda = 0, \quad \lambda \in \mathbb{R}\ \ \ \ (21)

\lim_{|\lambda| \to \infty}\chi^\pm(x, \lambda) = 1\!\!1\ \ \ \ (22),

where G(\lambda) depends on the so called scattering data:

G(\lambda) = \frac{1}{a^-(\lambda)} \begin{bmatrix} 1 & b^-(\lambda) \\ b^+(\lambda) & 1 \end{bmatrix}.

The main goal for the dressing method is to create new solution starting from a known FAS \chi^\pm_0(x, \lambda) and the Lax system with potential q^{0}. The new solutions will correspond to a new potential q^{1} with corresponding additional pair of discrete eigenvalues \lambda_1^\pm \in \mathbb{C}^{\pm} which are assumed to be simple.
The new and old solutions are related with a dressing factor g(x, \lambda) which we will find later, as follows:

\chi^+_1(x, \lambda) = g(x, \lambda)\chi^+_0(x, \lambda). \ \ \ \ (23)

q^{0} and q^{1} satisfy the spectral problem:

(i\frac{d}{dx} + q^0(x) - \lambda \sigma_3)\chi_0(x, \lambda) = 0\ \ \ \ (24),

(i\frac{d}{dx} + q^1(x) - \lambda \sigma_3)g\chi_0(x, \lambda) = 0\ \ \ \ (25).

From (25) we have

(i\partial_{x} g)\chi_0 + g i (\partial_{x} \chi_0) + (q^{1} - \lambda \sigma_{3})g \chi_{0} = 0

and from (24)

(i\partial_{x} g)\chi_{0} + g [-(q^{0} - \lambda \sigma_{3})] + (q^{1} - \lambda \sigma_{3})g \chi_{0} = 0

which we can state in the form of the following lemma:

Lemma 1.
i\partial_x g + q^1g - gq^0 + \lambda[g, \sigma_3] = 0.

Lemma 2.
For g, we use ansatz g(x, \lambda) = 1 + \frac{\lambda_{1}^{-} - \lambda_1^{+}}{\lambda - \lambda_{1}^{-}}P(x), where P will be determined in what follows.
Now if we insert g from Lemma 2 into expression in Lemma 1, we get

i\partial_{x} \frac{\lambda_{1}^{-} - \lambda_{1}^{+}}{\lambda - \lambda_{1}^{-}}P(x) + q^{1}(x)\left(1 + \frac{\lambda_{1}^{-} - \lambda_{1}^{+}}{\lambda- \lambda_{1}^{-}}P(x)\right) - \left(1 + \frac{\lambda_{1}^{-} - \lambda_1^{+}}{\lambda - \lambda_1^{-}}P(x)\right)q^{0}(x) + \\ + \lambda\left[\frac{\lambda_{1}^{-} - \lambda_{1}^{+}}{\lambda - \lambda_{1}^{-}}P(x), \sigma_{3} \right] = 0\ \ \ \ (26)

where

\left[\frac{\lambda_1^- - \lambda_1^+}{\lambda - \lambda_1^-}P(x)_, \sigma_3 \right]

is a commutator.
Since (26) must hold true for  \lambda \to \infty and \lambda \to \lambda_{1}^{-}, when \lambda \to \infty follows

q^{1}(x) - q^{0}(x) = -(\lambda_{1}^{-} - \lambda_{1}^{+}) \big[ P, \sigma_{3} \big]\ \ \ \ (27)

and when \lambda \to \lambda_{1}^{-} we get

\lim_{\lambda \to \lambda_{1}^{-}} \frac{\lambda_{1}^{-} - \lambda_{1}^{+}}{\lambda- \lambda_{1}^{-}}\left(i\partial_{x}P(x) + q^{1}(x)P(x) - P(x)q^{0}(x) + \\ + \lambda[P, \sigma_{3}]\right) + q^{1}(x) - q^{0}(x) = 0

which will be zero only if

i\partial_{x}P(x) + q^{1}(x)P(x) - P(x)q^{0}(x) + \lambda_{1}^{-}[P, \sigma_{3}] = 0.\ \ \ \ (28)

If we substitute (27) into (28)

i\partial_xP(x) + \left(q^0(x) - (\lambda_1^- - \lambda_1^+) \big[ P, \sigma_3 \big]\right)P(x) - P(x)q^0(x) + \lambda_1^-[P, \sigma_3] = 0\ \ \ \ (29)

or, after some calculations

i\partial_{x}P(x) + q^0(x)P(x) - P(x)q^0(x) + (\lambda_1^+ - \lambda_1^-) P(x) \sigma_3 P(x) +\\ +\lambda_1^- P(x)\sigma_3 - \lambda_1^+ \sigma_3 P(x) = 0.\ \ \ \ (30)

Proposition.
Projector P defined as

P := \frac{|n_{1}><m_{1}|}{<m_{1}|n_{1}>}\ \ \ \ (31)

where |n_{1}> is a two component column vector and <m_{1}| is a two component row vector satisfying

i\partial_x |n_{1}(x, t)> + (q^0 - \lambda_1^+ \sigma_3) |n_{1}(x, t)> = 0 \\ \implies |n_{1}(x, t)> = \chi_0(\lambda_1^{+}, x, t)|n_{0}>,

and

i\partial_x <m_{1}(x, t)| - <m_{1}(x, t)|(q^0 - \lambda_1^- \sigma_3) = 0 \\ \implies <m_{1}(x, t)| = <m_{0}| \chi_0^{-1}(\lambda_1^{-}, x, t).

is a solution of (30).
In this proposition we are using a bra-ket notation as a convention from physics, <m| = |m>^{\tau}.
Proof

Projector has the property P^{2} = P and we can see that this property will be satisfied if we substitute (31) in it.

Let us show the following:

i \partial_x \chi^{-1} = \chi^{-1}(q - \lambda \sigma_3).

First, we know that

i \partial_x\left( \chi \chi^{-1} \right) = 0

since

\chi \chi^{-1}

is identity.
After taking the derivative we get,

\left(i \partial_x \chi \right)\chi^{-1} + \chi i \partial_x \chi^{-1} = 0

or

-(q - \lambda \sigma_3) \chi \chi^{-1} + \chi i \partial_x \chi^{-1} = 0.

From here follows, after applying \chi^{-1}

i \partial_x \chi^{-1} = \chi^{-1}(q - \lambda \sigma_3).\ \ \ \ (32)

Now, if we take

|n_{1}(x)> = \chi(\lambda_1^+) |n_{0}(x)>

we get

i\partial_x |n_{1}(x)> + (q - \lambda_1^+ \sigma_3) |n_{1}(x)> = 0,

and from

<m_{1}(x)| = <m_{0}(x)| \chi^{-1}(\lambda_1^-)

follows

i\partial_x <m_{1}(x)| - <m_{1}(x)|(q - \lambda_1^- \sigma_3) = 0.

So P(x) satisfies (30) provided that the eigenvectors  <m_{1}(x)| and |n_{1}(x)> are solutions of the equations

\chi |n_{1}(x)> = i \partial_x |n_{1}(x)> + (q_0(x) -\lambda_1^+ \sigma_3)|n_{1}(x)> = 0\ \ \ \ (33)

<m_{1}(x)|\chi^{-1} = i \partial_x<m_{1}(x)| - <m_{1}(x)|(q_0(x) -\lambda_1^- \sigma_3) = 0\ \ \ \ (34)

Let us show that P(x) satisfies (30). If we insert (31) into (30), we get

i \partial_x \left( \frac{|n_{1}><m_{1}|}{<m_{1}|n_{1}>} \right) + q^0(x) \left( \frac{|n_{1}><m_{1}|}{<m_{1}|n_{1}>} \right) - \left( \frac{|n_{1}><m_{1}|}{<m_{1}|n_{1}>} \right)q^0(x) + \\ + (\lambda_1^+ - \lambda_1^-)\left( \frac{|n_{1}><m_{1}|}{<m_{1}|n_{1}>} \right)\sigma_3\left( \frac{|n_{1}><m_{1}|}{<m_{1}|n_{1}>} \right) + \\ + \lambda_1^{-} \left( \frac{|n_{1}><m_{1}|}{<m_{1}|n_{1}>} \right) \sigma_3 - \lambda_1^{+} \sigma_3 \left( \frac{|n_{1}><m_{1}|}{<m_{1}|n_{1}>} \right) = 0.\ \ \ \ (35)

After applying the derivative, taking into account (33), (34) all terms will cancel and that the expression will indeed equal zero.

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].