Kundu equation

Quantum Aspects

Though the Kundu-Ekchaus equation (3) is gauge equivalent to the

${\displaystyle {H}=\int dx\left[:\left((\psi _{x}^{\dagger }\psi _{x}+c\rho ^{2}+i\kappa \rho (\psi ^{\dagger }\psi _{x}-\psi _{x}^{\dagger }\psi )\right):+\kappa ^{2}(\psi ^{\dagger }\rho ^{2}\psi )\right],\ \ \ \ \rho \equiv (\psi ^{\dagger }\psi )}$

and defined through the

${\displaystyle [\psi (x),\psi ^{\dagger }(y)]=\delta (x-y)}$

${\displaystyle \ :\ \ :\ }$

${\displaystyle \delta }$

One-dimensional Anion gas

However, under a nonlinear transformation of the field below:

${\displaystyle {\tilde {\psi }}(x)=e^{-i\kappa \int _{-\infty }^{x}\psi ^{\dagger }(x')\psi (x')dx'}\psi (x)}$

the model can be transformed to:

${\displaystyle {\tilde {H}}=\int dx\vdots \left({\tilde {\psi }}_{x}^{\dagger }{\tilde {\psi }}_{x}+c({\tilde {\psi }}^{\dagger }{\tilde {\psi }})^{2}\right)\vdots ,}$

i.e. in the same form as the quantum model of the

${\displaystyle {\tilde {\psi }}^{\dagger }(x_{1}){\tilde {\psi }}^{\dagger }(x_{2})=e^{i\kappa \epsilon (x_{1}-x_{2})}{\tilde {\psi }}^{\dagger }(x_{2}){\tilde {\psi }}^{\dagger }(x_{1}),\ {\tilde {\psi }}(x_{1}){\tilde {\psi }}^{\dagger }(x_{2})=e^{-i\kappa \epsilon (x_{1}-x_{2})}{\tilde {\psi }}^{\dagger }(x_{2}){\tilde {\psi }}(x_{1})+\delta (x_{1}-x_{2})}$

etc. where

${\displaystyle \epsilon (x-y)=+\ ,-,0\ \ ~}$ for ${\displaystyle \ ~x>y,\ x<y,\ \ x=y,}$

though at the coinciding points the bosonic

${\displaystyle \delta }$