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.