Kundu equation

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

with arbitrary function

θ

(

t

,

x

)

{\displaystyle \theta (t,x)}

and the subscripts denoting partial derivatives. Equation (1) is shown to be reducible for the choice of

θ

x

=

−

κ

|

q

|

2

{\displaystyle \theta _{x}=-\kappa |q|^{2}}

to an integrable class of mixed nonlinear Schrödinger equation with cubic–quintic nonlinearity, given in a representative form

Here

α

,

c

,

κ

{\displaystyle \alpha ,c,\kappa }

are independent parameters, while

γ

=

κ

(

4

κ

+

α

)

.

{\displaystyle \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 solutions.

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

Kundu-Eckhaus equation

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

which may be obtained from the Kundu Equation (2), when restricted to

α

=

0

{\displaystyle \alpha =0}

. The same equation, limited further to the particular case

c

=

0

,

{\displaystyle c=0,}

was introduced later as the Eckhaus equation, following which equation (3) is presently known as the Kundu-Ekchaus equation.