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 in the form

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

which may be obtained from the Kundu Equation (2), when restricted to <math>\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 , 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 , <ref name={{Ohta|2012}} > Ohta, Y.; Yang, J. (2012), "General higher order rogue waves and their dynamics in the NLS equation", Proc. Royal Soc. Lond. A, 468: 1716