Darboux Transformations

Please see the paper to review the theory of Darboux Trasformations, and the Darboux Transformation Examples page for examples.

Supported Equations

The most general equation supported by the Darboux transformation in NonlinearSchrodinger.jl is of the following form:

\[i{\psi _x} + S[\psi (x,t)] - i\alpha H[\psi (x,t)] + \gamma P[\psi (x,t)] - i\delta Q[\psi (x,t)] = 0,\]

where

\[\begin{aligned} S[\psi (x,t)] &= \frac{1}{2}{\psi _{tt}} + {\left| \psi \right|^2}\psi, \\ H[\psi (x,t)] &= {\psi _{ttt}} + 6{\left| \psi \right|^2}{\psi _t}, \\ P[\psi (x,t)] &= {\psi _{tttt}} + 8{\left| \psi \right|^2}{\psi _{tt}} + 6{\left| \psi \right|^4}\psi + 4{\left| {{\psi _t}} \right|^2}\psi + 6{\psi _t}^2{\psi ^*} + 2{\psi ^2}\psi _{tt}^*, \\ Q[\psi (x,t)] &= {\psi _{ttttt}} + 10{\left| \psi \right|^2}{\psi _{ttt}} + 30{\left| \psi \right|^4}{\psi _t} + 10\psi {\psi _t}\psi _{tt}^* + 10\psi \psi _t^*{\psi _{tt}} + 20{\psi ^*}{\psi _t}{\psi _{tt}} + 10\psi _t^2\psi _t^*. \end{aligned}\]

Special cases include the cubic nonlinear Schrodinger equation ($\alpha = \gamma = \delta = 0$), the Hirota equation ($\alpha \neq 0, \gamma = \delta = 0$) the Lakshmanan-Porsezian-Daniel (LPD) equation ($\gamma \neq 0, \alpha = \delta = 0$) and the Quintic nonlinear Schrodinger equation ($\delta \neq 0, \alpha = \gamma = 0$).

For this generalized NLSE, we support both the breather and soliton seeds. Additionally, for the cubic NLSE, we support the $cn$ and $dn$ seeds. Support for these seeds will be added at some point in the future for the generalized NLSE.