Weikang Xun | Searcharxiv

arXiv2022

Painleve-type asymptotics for the defocusing Hirota equation in transition region

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 long-time asymptotics of the solution in the critical transition region $|\frac{x}{t} - \frac{α^2}{3β}|t^{2/3} \leq M$, $M$ 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 $\mathrm{II}$ equation. Keywords: Hirota equation, steepest descent method, Painlevé $\mathrm{II}$ equation, long-time asymptotics.

math-ph↗

arXiv2021

Long time and Painleve-type asymptotics for the Sasa-Satsuma equation in solitonic space time regions

The Sasa-Satsuma equation with $3 \times 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 $\overline{\partial}$-nonlinear steepest descent method, we find qualitatively different long time asymptotic forms for the Sasa-Satsuma equation in three solitonic space-time regions: (1)\ For the region $x<0, |x/t|=\mathcal{O}(1)$, the long time asymptotic is given by $$q(x,t)=u_{sol}(x,t| σ_{d}(\mathcal{I})) + t^{-1/2} h + \mathcal{O} (t^{-3/4}). $$ in which the leading term is $N(I)$ solitons, the second term the second $t^{-1/2}$ order term is soliton-radiation interactions and the third term is a residual error from a $\overline\partial$ equation. (2)\ For the region $ x>0, |x/t|=\mathcal{O}(1)$, the long time asymptotic is given by $$ u(x,t)= u_{sol}(x,t| σ_{d}(\mathcal{I})) + \mathcal{O}(t^{-1}).$$ in which the leading term is $N(I)$ solitons, the second term is a residual error from a $\overline\partial$ equation. (3) \ For the region $ |x/t^{1/3}|=\mathcal{O}(1)$, the Painleve asymptotic is found by $$ u(x,t)= \frac{1}{t^{1/3}} u_{P} \left(\frac{x}{t^{1/3}} \right) + \mathcal{O} \left(t^{2/(3p)-1/2} \right), \qquad 4<p < \infty.$$ in which the leading term is a solution to a modified Painleve $\mathrm{II}$ equation, the second term is a residual error from a $\overline\partial$ equation.

math.AP↗