We study the effect of curvature and torsion on the solitons of the nonlinear Dirac equation considered on planar and space curves. Since the spin connection is zero for the curves considered here, the arc variable provides a natural setting to understand the role of curvature and then we can obtain the transformation for the 1+1 dimensional Dirac equation directly from the metric. Depending on the curvature, the soliton profile either narrows or expands. Our results may be applicable to yet-to-be-synthesized curved quasi-one dimensional Bose condensates.

中文翻译：

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约束到平面和空间曲线的 2+1 和 3+1 维非线性狄拉克方程的孤立波解

我们研究曲率和扭转对平面和空间曲线上考虑的非线性狄拉克方程的孤子的影响。由于这里考虑的曲线的自旋连接为零，弧变量提供了一个自然的设置来理解曲率的作用，然后我们可以直接从度量中获得 1+1 维狄拉克方程的变换。根据曲率，孤子轮廓要么变窄，要么扩展。我们的结果可能适用于尚未合成的弯曲准一维玻色凝聚。