User:Anjan.kundu/sandbox

Kundu-Eckhaus Equation
An integrable generalization of  nonlinear Schroedinger equation with additional quintic nonlinerity and a nonlinear dispersive term was proposed in in the form
 * $$ \psi_t+\psi_{xx} -2  c |\psi|^2q+\kappa^2|\psi|^4\psi \pm 2i

\kappa (| \psi|^2)_x \psi=0,  .... (1) $$ was proposed  in which may be derived  from the Kundu Equation, when restricted to $$\alpha =0$$. The same equation, limited further to the particular case $$c =0,$$ was introduced later as Eckhaus equation, following which  equation (3) is known as the   Kundu-Ekchaus eqution. The  Kundu-Ekchaus equation  can be reduced to the nonlinear Schroedinger equation through a nonlinear transformation of the field and known therefore to be gauge equivalent integrable systems, since they are equivalent under the gauge transformation.

Properties and Applications
the  Kundu-Ekchaus equation is asociated with a Lax pair, higher conserved quantity, exact soliton solution, rogue wave solution etc. Over the years various aspects of this equation, its generalizations and link with other equations have been studied. In particular, relationship of   Kundu-Ekchaus equation with the Johnson's hydrodynamic equation near criticality is established,  its  discretizations ,  reduction via Lie symmetry  , complex structure via Bernoulli subequation , bright and dark soliton solutions] via [[Baecklund transfomation and Darbaux transformation with the associated rogue wave solutions, are studied. RKL equation

A multi-component generalisation of the   Kundu-Ekchaus equation (3), known as Radhakrishnan, Kundu and Laskshmanan (RKL) equation was proposed in nolinear optics for fiber communication through solitonic pulses in a birefringent non-Kerr medium and analysed subsequently for its exact soliton solution and other aspects in a series of papers

Quantum Aspect
Though the  Kundu-Ekchaus equation (3) is gauge equivalent to the nonlinear Schroedinger equation, they differ in an interesting way with respect to their Hamiltinian structures and field commutation relations. The Hamiltonian   operator of the   Kundu-Ekchaus equation quantum field  model given by
 * $$ {H}

=\int dx \left[ : \left( (\psi^\dagger_x \psi_x + c  \rho^2 +i \kappa \rho (\psi^\dagger \psi_x- \psi^\dagger_x \psi) \right): +\kappa^2 (\psi^\dagger \rho ^2 \psi) \right], \ \ \ \ \rho \equiv (\psi^\dagger \psi) $$ and defined through  the  bosonic  field  operatorcommutation relation  $$ [\psi (x), \psi^\dagger(y)]= \delta(x-y)$$,  is more complicated than the well known bosonic   Hamiltonian of the quantum nonlinear Schroedinger equation. Here $$\ : \ \ : \ $$  indicates normal ordering  in bosonic operators. This model  corresponds to a double $\delta $ function  interacting bose gas and difficult to solve directly.

 one-dimensional Anyon gas

However under a nonlinear  transformation of the field

\tilde \psi (x)= e^{-i \kappa \int^x_{- \infty} \psi^\dagger (x') \psi (x') dx'} \psi (x) $$ the model can be transformed to

\tilde H=\int dx \vdots \left( \tilde \psi^\dagger_x \tilde \psi_x + c (\tilde \psi^\dagger \tilde \psi)^2 \right) \vdots , $$ i.e. in the same form as the quantum model of nonlinear Schroedinger equation (NLSE), though   it differs from the NLSE in its contents , since now the fields involved are no longer bosonic operators but exhibit anyon like properties

\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

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

though at the coinciding poiints the bosonic commutation relation still holds. In analogy with the Lieb Limiger model of $$ \delta $$ function  bose gas, the quantum Kundu-Ekchaus  model in the N-particle sector therefore corresponds to an  one-dimensional (1D) anyon gas interacting via a $$  \delta  $$ function interaction. This model of  interacting anyon gas was proposed and eaxctly solved by the Bethe ansatz in and  this basic  anyon model is studied further for investigating  various aspects of the  1D anyon gas as well as extended in different directions