<p style='text-indent:20px;'>In this work, a generalized nonlocal Lakshmanan-Porsezian-Daniel (LPD) equation is introduced, and its integrability as an infinite dimensional Hamilton dynamic system is established. We successfully derive the inverse scattering transform (IST) of the nonlocal LPD equation. The direct scattering problem of the equation is first constructed, and some important symmetries of the eigenfunctions and the scattering data are discussed. By using a novel Left-Right Riemann-Hilbert (RH) problem, the inverse scattering problem is analyzed, and the potential function is recovered. By introducing the special conditions of reflectionless case, the time-periodic soliton solutions formula of the equation is derived successfully. Take <inline-formula><tex-math id="M1">\begin{document}$ J = \overline{J} = 1,2,3 $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M2">\begin{document}$ 4 $\end{document}</tex-math></inline-formula> for example, we obtain some interesting phenomenon such as breather-type solitons, arc solitons, three soliton and four soliton. Furthermore, the influence of parameter <inline-formula><tex-math id="M3">\begin{document}$ \delta $\end{document}</tex-math></inline-formula> on these solutions is further considered via the graphical analysis. Finally, the eigenvalues and conserved quantities are investigated under a few special initial conditions.</p>
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.
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.
The Sasa-Satsuma equation with 3 × 3 Lax representation is one of the integrable extensions of the nonlinear Schrödinger equation. In this paper, we consider the Cauchy problem of the Sasa-Satsuma equation with generic decaying initial data. Based on the Rieamnn-Hilbert problem characterization for the Cauchy problem and the ∂-nonlinear steepest de-
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