一维和二维二分量离散非线性薛定谔晶格上的非交错-交错孤子
Unstaggered-staggered solitons on one- and two-dimensional two-component discrete nonlinear Schrödinger lattices
摘要：
本文研究了一维和二维格点域上两个耦合的离散非线性薛定谔方程组中产生的耦合非交错-交错孤子对，方程组中的非线性薛定谔方程组具有自吸引的原位自相位调制非线性，并通过排斥的交叉相位调制相互作用耦合.这些混合模式属于“共生”类型，因为每个孤立的分量可能只携带普通的非交错孤立子。虽然DNLS系统的大多数工作都是针对对称的以站点为中心的基本孤子，但这些模型产生了各种其他激发态，这些激发态也可能是稳定的。其中最简单的是离散扭曲孤子形式的反对称态，它在连续极限中没有对应物。在二维晶格域的扩展中，扭曲态的自然对应物是涡旋孤子。我们首先介绍了变分近似（VA）的孤子，然后对其进行数值校正，以构建精确的定态解，然后将其用作模拟的初始条件，以检查定态在时间演化下是否持续。所得到的双分量解包括：（i）一维基-扭和扭-扭孤子对，（ii）二维基-基孤子对，（iii）二维涡旋-涡旋孤子对。我们还强调了各种其他瞬态动力学制度，如呼吸和振幅死亡。研究结果适用于模拟二元玻色-爱因斯坦凝聚体，加载在深晶格势，具有相同或不同的原子质量的两个组件，和阵列的双峰光波导。
英文摘要：
We study coupled unstaggered-staggered soliton pairs emergent from a system of two coupled discrete nonlinear Schrödinger (DNLS) equations with the self-attractive on-site self-phase-modulation nonlinearity, coupled by the repulsive cross-phase-modulation interaction, on 1D and 2D lattice domains. These mixed modes are of a "symbiotic" type, as each component in isolation may only carry ordinary unstaggered solitons. While most work on DNLS systems addressed symmetric on-site-centered fundamental solitons, these models give rise to a variety of other excited states, which may also be stable. The simplest among them are antisymmetric states in the form of discrete twisted solitons, which have no counterparts in the continuum limit. In the extension to 2D lattice domains, a natural counterpart of the twisted states are vortical solitons. We first introduce a variational approximation (VA) for the solitons, and then correct it numerically to construct exact stationary solutions, which are then used as initial conditions for simulations to check if the stationary states persist under time evolution. Two-component solutions obtained include (i) 1D fundamental-twisted and twisted-twisted soliton pairs, (ii) 2D fundamental-fundamental soliton pairs, and (iii) 2D vortical-vortical soliton pairs. We also highlight a variety of other transient dynamical regimes, such as breathers and amplitude death. The findings apply to modeling binary Bose-Einstein condensates, loaded in a deep lattice potential, with identical or different atomic masses of the two components, and arrays of bimodal optical waveguides.
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