Painlevé XXXIV asymptotics for the defocusing nonlinear Schrödinger equation with a finite-genus algebro-geometric background

Abstract

In this paper, we consider the Cauchy problem for the defocusing nonlinear Schr$\ddot{\text{o}}$dinger equation with a finite genus algebro-geometric background. Long-time asymptotics of the solution are derived in four space-time regions. It comes out that the leading-order term in the expansion is, up to a constant, given by the background solution with a shift of the parameter. The subleading term, however, decays at different rates for different regions. We particularly highlight that in the two transition regions, they are of order $\mathcal{O}(t^{-1/3})$ and the coefficients involve an integral of the Painlevé XXXIV transcendent. We establish our results by applying a nonlinear steepest descent analysis to the associated Riemann-Hilbert problems.

Figures (9)

• Figure 1: Four different asymptotic regions given in Definition \ref{['def:regions']} with $n=2$.
• Figure 2: The canonical homology basis $\{a_j,b_j\}_{j=1}^n$ on the Riemann surface $\mathcal{R}$. Here the solid and dashed arcs indicate the parts on the upper and lower sheet, respectively.
• Figure 3: Two examples for the signature table of $\mathrm{Im}\, \theta$: (a) $\xi\in(\xi_2,\hat{\xi}_1)$; (b) $\xi\in(\hat{\xi}_1,\xi_1)$. $\mathrm{Im}\, \theta =0$ on blue curve. The "+" represents where $\mathrm{Im}\, \theta>0$ and "-" represents where $\mathrm{Im}\, \theta<0$.
• Figure 4: The contours $\Sigma_i$ and the regions $\Omega_i$, $i=1,2$, in \ref{['def Omega sigma1']} and \ref{['def Omega sigma2']}.
• Figure 5: The contour $\Sigma^{(1)}\cap U$ in the $z$-plane and the contour $\Sigma^{(loc,\zeta)}\cap U^{(\zeta)}$ in the $\zeta$-plane under the mapping \ref{['def zeta1']}.

Theorems & Definitions (25)

• Definition 2.1
• Theorem 2.3
• Proposition 3.2
• proof
• Proposition 3.3
• Proposition 3.4
• proof
• Remark 3.5
• Proposition 3.8
• proof