Kundu equation

Not to be confused with Kundu–Eckhaus equation,(See below).

The Kundu equation is a general form of integrable system that is gauge-equivalent to the mixed nonlinear Schrödinger equation. It was proposed by Anjan Kundu. as

$iq_{t}+q_{xx}+i\alpha (|q|^{2}q)_{x}+c|q|^{2}q-(\theta _{t}+\theta _{x}^{2}-i\theta _{xx})q+\theta _{x}(2iq_{x}-\alpha |q|^{2}q)=0,\ \ ......(1)$

with arbitrary function $\theta (t,x)$ and the subscripts denoting partial derivatives. Equation (1) is shown to be reducible for the choice of $\theta _{x}=-\kappa |q|^{2}$ to an integrable class of mixed nonlinear Schrödinger equation with cubic–quintic nonlinearity,given in a representative form

$q_{t}+q_{xx}+i\alpha (|q|^{2}q)_{x}+c|q|^{2}q+\gamma |q|^{4}q+4i\kappa (|q|^{2})_{x}q=0.\ \ ......(2)$

Here $\alpha ,c,\kappa$ are independent parameters, while $\gamma =\kappa (4\kappa +\alpha ).$ Equation (1), more specifically equation (2) is known as the Kundu equation.

Properties and applications

The Kundu equation is a completely integrable system, allowing Lax pair representation, exact solutions and higher conserved quantity. Along with its different particular cases this equation has been investigated for finding its exact travelling wave solutions, exact solitary wave solutions via bilinearization, and Darboux transformation together with the orbital stability for such solitary wave and the related rogue wave solution. Painlevé analysis establishes the integrability for this family of nonlinear Schrödinger equations.

The Kundu equation was applied also to various physical processes like fluid dynamics, plasma physics, nonlinear optics etc. (See for relevant references) and is linked to mixed nonlinear Schrödinger equation through gauge transformation and reducible to a variety of known integrable equations like nonlinear Schrödinger equation (NLSE), derivative NLSE, higher nonlinear derivative NLSE, Chen–Lee–Liu, Gerjikov-Vanov, Kundu–Eckhaus equations etc.,for different choices of the parameters.

Kundu-Eckhaus Equation

A generalization of nonlinear Schroedinger equation with additional quintic nonlinerity and a nonlinear dispersive term was proposed in in the form

$\psi _{t}+\psi _{xx}-2c|\psi |^{2}q+\kappa ^{2}|\psi |^{4}\psi \pm 2i\kappa (|\psi |^{2})_{x}\psi =0,......(3)$

which may be obtained from the Kundu Equation (2), 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 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 , 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 optics communication through soliton 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

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

and defined through the 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

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

the model can be transformed to

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

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

External links

• Painleve Analysis.
• Darboux Transformation.
• Mixed Nonlinear Schrödinger equation
• Gerdjikov-Ivanov equation
• 1D Anyon
• How fiber optics works