We consider the Cauchy problem for the classical Hirota equation on the line with decaying initial data. Based on the spectral analysis of the Lax pair of the Hirota equation, we first expressed the solution of the Cauchy problem in terms of the solution of a Riemann-Hilbert problem. Further we apply nonlinear steepest descent analysis to obtain the longtime asymptotics of the solution in the critical transition region | x t − α 2 3β |t 2/3 ≤ M , M is a positive constant. Our result shows that the long time asymptotics of the Hirota equation can be expressed in terms of the solution of Painlevé II equation.
We consider the long-time asymptotics for the defocusing Hirota equation with Schwartz Cauchy data in the transition region. On the basis of direct and inverse scattering transform of the Lax pair of Hirota equations, we first express the solution of the Cauchy problem in terms of the solution of a Riemann–Hilbert problem. Further, we apply nonlinear steepest descent analysis to obtain the long-time asymptotics of the solution in the critical transition region
|
x
/
t
−
(
α
2
/
3
β
)
|
t
2
/
3
≤
M
, where
M
is a positive constant. Our result shows that the long-time asymptotics of the Hirota equation can be expressed in terms of the solution of the Painlevé
II
equation.
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