User:Anjan.kundu/sandbox
Kundu-Eckhaus Equation
A generalization of nonlinear Schroedinger equation with additional quintic nonlinerity and a nonlinear dispersive term was proposed in [1] in the form
which may be obtained from the Kundu Equation (2), when restricted to Failed to parse (syntax error): {\displaystyle \alpha =0<math>. The same equation, limited further to the particular case <math>\beta =0,<math> was introduced later as [[Eckhaus equation]], following which equation (3) is presently 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<ref name={{kundu|1987}} > {{citation | first=A. |last=Kundu |title= Exact solutions in higher order nonlinear equations gauge transformation | journal= Physica D | volume=25 | year=1987 |pages=399-406 }} </ref> , its [[discretizations]] <ref name={{levi|2009}} > {{citation | first1=D. |last1=Levi| first2=C. |last2=Scimiterna |title= The Kundu–Eckhaus equation and its discretizations | journal= J. Phys. A | volume= | year=2009 |pages= }} </ref> , reduction via [[Lie symmetry]] <ref name={{tooman|2013}} > {{citation | first1= |last1=Toomanian| first2= |last2=Asadi |title= Reductions for Kundu-Eckhaus equation via Lie symmetry analysis | journal= Math. Sciences | volume=7 | year=2013 |pages=50 }} </ref> , complex structure via [[Bernoulli subequation]] <ref name={{beok|2015}} > {{citation | first1=H. M. |last1= Beokonus| first2=Q. H. |last2= Bulut |title= On the complex structure of Kundu-Eckhaus equation via Bernoulli subequation fungtion method | journal= Waves in Random and Complex Media | volume= 28 Aug. | year=2015 |pages= }} </ref> , bright and dark [[soliton solutions] via [[Baecklund transfomation]] <ref name={{wang|2015}} > {{citation | first1=H. P. |last1= Wang,| first2= et. |last2= al. |title= Bright and Dark solitons and Baecklund transfomation for the Kundu–Eckhaus equation | journal= Appl. Math. Comp. | volume= 251 | year=2015 |pages= 233}} </ref> and [[Darbaux transformation]]<ref name={{Qui|2015}} >{{citation | first1= D. |last1= Qui| first2= et. |last2= al. |title=The Darbaux transformation and the Kundu–Eckhaus equation| journal= Proc. Royal Soc. Lond. A | volume= 451 | year=2015|pages= 20150236}}</ref> with the associated [[rogue wave]] solutions <ref name={{XWang|2014}} > {{citation | first1= Xin |last1= Wang| first2= et. |last2= al. |title= Higher-order rogue wave solutions of the Kundu–Eckhaus equation | journal= Phys. Scr. | volume= 89 | year=2014 |pages= 095210}} </ref> ,<ref name={{Ohta|2012}} > {{citation | first1= Y. |last1= Ohta| first2= J. |last2= Yang |title= General higher order rogue waves and their dynamics in the NLS equation | journal= Proc. Royal Soc. Lond. A | volume= 468 | year=2012 |pages= 1716}}</ref> 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]] <ref name={{RKL|1999}} > {{citation | first1= R. |last1= Radhakrishnan| first2= A. |last2= Kundu| first3= M. |last2= Lakshmanan|title= Coupled nonlinear Schr\"odinger equations with cubic-quintic nonlinearity: integrability and soliton interaction in non-Kerr media | journal= Phys. Rev. E | volume= 60| year=1999 |pages= 3314}} </ref> and analysed subsequently for its exact [[soliton solution]] and other aspects in a series of papers<ref name={{Biswas|2009}} > {{citation | first1= A. |last1= Biswas|title= 1-soliton solution of the generalized Radhakrishnan, Kundu, Lakshmanan equation | journal= Physics Letters A | volume= 373 | year=2009 |pages= 2546}} </ref> , <ref name={{Zhang|2008}} > {{citation | first1= J. L. |last1= Zhang| first2= M. L. |last2= Wang |title= Various exact solutions for two special type RKL models | journal= Chaos Solitons Fractals | volume= 37 | year=2008 |pages= 215}}</ref> ,<ref name={{ Ganji|2008}} > {{citation | first1= D. D. |last1= Ganji| first2= et. |last2= al. |title= Exp-function based solution of nonlinear Radhakrishnan, Kundu and Laskshmanan (RKL) equation | journal= Acta Appl. Math. | volume= 104 | year=2008 |pages= 201}} </ref> , <ref name={{ Ganji|2008}} > {{citation | first1= X. I. N. |last1= Chun-gang| first2= et. |last2= al. |title= New soliton solution of the generalized RKL equation through optical fiber transmission | journal= J. Anhui Univ. (Natural Sc Edition) | volume= 35 | year=2011 |pages= 39}} </ref> ===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 :<math> {H} =\int dx \left[ : \left( (\psi^\dag_x \psi_x + c \rho^2 +i \kap \rho (\psi^\dag \psi_x- \psi^\dag_x \psi) \right): +\kap^2 (\psi^\dag \rho ^2 \psi) \right], \ \ \ \ \rho \equiv (\psi^\dag \psi) } and defined through the bosonic field operatorcommutation relation Failed to parse (unknown function "\dag"): {\displaystyle [\psi (x), \psi^\dag(y)]= \de(x-y)} , is more complicated than the well known bosonic Hamiltonian of the quantum [[nonlinear Schroedinger equation]]. Here Failed to parse (unknown function "\dag"): {\displaystyle \ : \ \ : \ <math> 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 :<math> \tilde \psi (x)= e^{-i \delta \int^x_{- \infty} \psi^\dag (x') \psi (x') dx'} \psi (x) } the model can be transformed to
- Failed to parse (unknown function "\dag"): {\displaystyle \tilde H=\int dx \vdots \left( \tilde \psi^\dag_x \tilde \psi_x + \beta (\tilde \psi^\dag \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
- Failed to parse (unknown function "\dag"): {\displaystyle \tilde \psi^\dag (x_1) \tilde \psi^\dag (x_2)=e^{i \delta \ep (x_1-x_2)} \tilde \psi^\dag (x_2)\tilde \psi^\dag (x_1) , \ \tilde \psi (x_1) \tilde \psi^\dag (x_2)=e^{-i \kap \ep (x_1-x_2)} , }
- Failed to parse (unknown function "\dag"): {\displaystyle \tilde \psi^\dag (x_2)\tilde \psi (x_1)+ \de (x_1-x_2) }
etc. where
- Failed to parse (syntax error): {\displaystyle \epsilon (x-y)= \pm 1 \ \mbox{for}\ x >y, \ x< y \ \mbox{and} \ =0 \ \mbox{for} \ 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 [2]