We address dissipative soliton formation in modulated parity-time ($\mathcal{PT}$)-symmetric continuous waveguide arrays composed from waveguides with amplifying and absorbing sections, whose density gradually increases (due to decreasing waveguide separation) either towards the center of the array or towards its edges. In such a structure, the level of gain/loss at which $\mathcal{PT}$-symmetry gets broken depends on the direction of increase of waveguide density. Breakup of $\mathcal{PT}$-symmetry occurs when eigenvalues of modes localized in the region where waveguide density is largest collide and move into a complex plane. In this regime of broken symmetry, the inclusion of focusing Kerr-type nonlinearity of the material and weak two-photon absorption allows to arrest the growth of amplitude of amplified modes and may lead to the appearance of stable attractors either in the center or at the edge of the waveguide array, depending on the type of array modulation. Such solitons can be stable; they acquire specific triangular shapes and notably broaden with increase of gain/loss level. Our results illustrate how spatial array modulation that breaks $\mathcal{PT}$-symmetry “locally” can be used to control the specific location of dissipative solitons forming in the array.

中文翻译：

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调制中的边缘散逸孤子 $\mathcal{P}\mathcal{Ť}$对称波导阵列

我们解决了在调制的奇偶校验时间内耗散孤子形成的问题（$\mathcal{PT}$对称的连续波导阵列，由具有放大和吸收部分的波导组成，其密度朝着阵列的中心或其边缘逐渐增加（由于减小的波导间隔）。在这样的结构中，$\mathcal{PT}$对称性的破坏取决于波导密度增加的方向。分手$\mathcal{PT}$当位于波导密度最大的区域中的模的特征值发生碰撞并移动到复杂平面中时，就会发生对称。在这种破坏对称的状态下，材料的聚焦Kerr型非线性和弱的双光子吸收的结合可阻止放大模式振幅的增长，并可能导致在中心或中心出现稳定的吸引子。波导阵列的边缘，取决于阵列调制的类型。这样的孤子可能是稳定的。它们具有特定的三角形形状，并且随着增益/损耗水平的增加而显着变宽。我们的结果说明了如何打破空间阵列调制$\mathcal{PT}$“局部”对称可用于控制阵列中形成的耗散孤子的特定位置。