Lax Equation formulation of the Focusing Non-Linear Schrodinger Equation.

Question

For a project I am writing at university, I have a section where I am writing about rational solutions to the NLSE, which use Lax pairs as the starting point for the derivation. I therefore wish to derive the Lax pair for the NLSE from first principles. In a paper I have been reading, the $\bf{KdV}$ equation was expressed in terms of the linear operators $\mathbf{L}$ and and $\mathbf{B}$ as follows: $$\frac{d\mathbf{L}}{dt} + [\mathbf{L},\mathbf{B}] = 0$$ where $$\mathbf{L} = -6\frac{d^{2}}{dx^{2}}$$ and $$\mathbf{B} = -4\frac{d^{3}}{dx^{3}} - u\frac{d}{dx} - \frac{1}{2}u_{x}.$$ I have been seraching around for a simmilar formulation for the focusing NLSE and have not been able to come up with anything. I am therefore wondering whether anyone could show me what the Lax equation formulation of the NLSE is and how I might go about deriving it.

Answer 1

The Lax pair of the Nonlinear Schrödinger equation was first given in 1972 by Zakharov and Shabat – see Equations (4) and (5) in the PDF of their paper https://www.semanticscholar.org/paper/Exact-Theory-of-Two-dimensional-Self-focusing-and-Zakharov-Shabat/73ef172d46291e1b2235d4fea6d80078f5e705ef?p2df

The Lax pair in this case is a $2\times 2$ matrix linear system, rather than a pair of scalar differential equations as in the case of the KdV.