Higher Order Asymptotics of the Modified Non-Linear Schrödinger Equation
Abstract
Using the matrix Riemann-Hilbert factorisation approach for non-linear evolution systems which take the form of Lax-pair isospectral deformations, the higher order asymptotics as
t→±∞
(x/t∼O(1))
of the solution to the Cauchy problem for the modified non-linear Schrödinger equation,
i
∂
t
u+1/2
∂
2
x
u+|u
|
2
u+is
∂
x
(|u
|
2
u)=0
,
s∈
R
>0
, which is a model for non-linear pulse propagation in optical fibres in the subpicosecond time scale, are obtained: also derived are analogous results for two gauge-equivalent non-linear evolution equations; in particular, the derivative non-linear Schrödinger equation,
i
∂
t
q+
∂
2
x
q−i
∂
x
(|q
|
2
q)=0
.