Inverse scattering theory for the discrete PT-symmetric nonlocal nonlinear Schröinger equation under arbitrarily large nonzero boundary conditions

Abstract

In this paper, the theory of inverse scattering transform (IST) is developed for the discrete PT-symmetric nonlocal nonlinear Schröinger equation under large nonzero boundary conditions (NZBCs). By considering that the data at infinity have constant amplitudes, two cases are studied where the previous IST theory fails for large NZBCs. Based on a suitable uniformization variable, the rigorous proofs for the analyticity, symmetries and asymptotic behaviors of the eigenfunctions and scattering coefficients are provided for the direc problem, and the potential reconstruction formula is derived by solving the Riemann-Hilbert problem. Particularly, the focusing equation is found to admit two types of novel solitons under large NZBCs: oscillating soliton and breather, where the former has not been previously reported, while the latter does not occur under small NZBCs. In addition, the multi-soliton solutions are shown to exhibit the collisions among oscillating dark/anti-dark solitons, and the superposition of oscillating soliton and breather.

Figures (5)

• Figure 1: The complex $z$-plane, including four branch points $z = \mathrm{i} r \pm \mathrm{i} Q_0, - \mathrm{i} r \pm \mathrm{i} Q_0$ and branch cut $(- \mathrm{i} r - \mathrm{i} Q_0, \mathrm{i} r - \mathrm{i} Q_0) \cup (- \mathrm{i} r + \mathrm{i} Q_0, \mathrm{i} r + \mathrm{i} Q_0)$.
• Figure 2: The regions $D^+$ (gray) and $D^-$ (white) in the complex $\zeta$-plane, which correspond to $|\lambda| < 1$ and $|\lambda| > 1$ in the complex $\lambda$-plane. The symmetries of discrete eigenvalues are also displayed.
• Figure 3: (a) Distribution of discrete eigenvalues $\{\zeta_{1}, \hat{\zeta}_{1}\}$; (b) An oscillating anti-dark soliton with $Q_+ = 2 e^{\frac{\pi}{3} \mathrm{i}}$, $\zeta_{1} = \frac{\sqrt{3} \mathrm{i} }{3}$, $\hat{\zeta}_{1} = -\sqrt{3} \mathrm{i}$ and $\theta_{1} = 2$. (c) An oscillating dark soliton with $Q_+ = 2 e^{\frac{\pi}{3}\mathrm{i}}$, $\zeta_{1} = \frac{\sqrt{3} \mathrm{i} }{3}$, $\hat{\zeta}_{1} = \sqrt{3} \mathrm{i}$ and $\theta_{1} = 0$.
• Figure 4: (a) Distribution of discrete eigenvalues $\{\zeta_{1}, \hat{\zeta}_{1}\}$, $\{\zeta_{2}, \hat{\zeta}_{2}\}$ and $\{\zeta_{3}, \hat{\zeta}_{3}\}$; (b) An interaction among three oscillating solitons with $Q_+ = 2 e^{\mathrm{i}}$, $\zeta_{1} = \frac{\sqrt{3} \mathrm{i} }{3}$, $\zeta_{2} = \frac{8 \sqrt{3} \mathrm{i} }{15} + \frac{ \sqrt{3} \mathrm{i} }{15}$, $\zeta_{3} = - \frac{8 \sqrt{3} \mathrm{i} }{15} + \frac{ \sqrt{3} \mathrm{i} }{15}$, $\hat{\zeta}_{1} = -\sqrt{3} \mathrm{i}$, $\hat{\zeta}_{2} = - \frac{8 \sqrt{3} \mathrm{i} }{13} - \frac{ \sqrt{3} \mathrm{i} }{13}$, $\hat{\zeta}_{3} = \frac{8 \sqrt{3} \mathrm{i} }{13} - \frac{ \sqrt{3} \mathrm{i} }{13}$, $\theta_{1} = 3$, and $\theta_{2} =\theta_{3} = 2$; (c) An interaction between two oscillating solitons with $Q_+ = 2 e^{\frac{\pi}{3} \mathrm{i} }$, $\zeta_{1} = \frac{\sqrt{3} \mathrm{i} }{3}$, $\zeta_{2} = \frac{8 \sqrt{3} \mathrm{i} }{15} + \frac{ \sqrt{3} \mathrm{i} }{15}$, $\zeta_{3} = - \frac{8 \sqrt{3} \mathrm{i} }{15} + \frac{ \sqrt{3} \mathrm{i} }{15}$, $\hat{\zeta}_{1} = -\sqrt{3} \mathrm{i}$, $\hat{\zeta}_{2} = - \frac{8 \sqrt{3} \mathrm{i} }{13} - \frac{ \sqrt{3} \mathrm{i} }{13}$, $\hat{\zeta}_{3} = \frac{8 \sqrt{3} \mathrm{i} }{13} - \frac{ \sqrt{3} \mathrm{i} }{13}$, $\theta_{1} = \frac{\pi}{3}$, and $\theta_{2} = \theta_{3} = -1$.
• Figure 5: (a) Distribution of discrete eigenvalues $\{\zeta_{1}, \hat{\zeta}_{1}\}$, $\{\zeta_{2}, \hat{\zeta}_{2}\}$ and $\{\zeta_{3}, \hat{\zeta}_{3}\}$; (b) Superposition of one oscillating dark soliton and one breather with $Q_+ = 2 e^{\frac{\pi}{3} \mathrm{i} }$, $\zeta_{1} = \frac{\sqrt{3} \mathrm{i} }{3}$, $\zeta_{2} = \frac{\sqrt{3} \mathrm{i} }{6}$, $\zeta_{3} = \frac{5\sqrt{3} \mathrm{i} }{9}$, $\hat{\zeta}_{1} = -\sqrt{3} \mathrm{i}$, $\hat{\zeta}_{2} = - \frac{3\sqrt{3} \mathrm{i} }{5}$, $\hat{\zeta}_{3} = -2 \sqrt{3} \mathrm{i}$, $\theta_{1} = 0$, $\theta_{2} = \frac{\pi}{4}$ and $\theta_{3} = \frac{\pi}{4}$; (c) One breather with $Q_+ = 2 e^{\frac{\pi}{4} \mathrm{i}}$, $\zeta_{1} = \frac{\sqrt{3} \mathrm{i} }{3}$, $\zeta_{2} = \frac{\sqrt{3} \mathrm{i} }{6}$, $\zeta_{3} = \frac{5\sqrt{3} \mathrm{i} }{9}$, $\hat{\zeta}_{1} = -\sqrt{3} \mathrm{i}$, $\hat{\zeta}_{2} = - \frac{3\sqrt{3} \mathrm{i} }{5}$, $\hat{\zeta}_{3} = -2 \sqrt{3} \mathrm{i}$, $\theta_{1} = \frac{\pi}{4}$, $\theta_{2} = 2$ and $\theta_{3} = 2$.

Theorems & Definitions (8)

• Theorem 2.1
• Theorem 2.2
• Lemma 2.3
• Lemma 2.4
• Lemma 3.1
• Lemma 3.2
• Theorem 4.1
• Remark 4.2