This article investigates the optical solitons to the three coupled Gross–Pitaevskii (GP) system (also called the non-linear Schrödinger (NLS) equation), which describes the $F=1$ spinor Bose–Einstein condensate, with $F$ denoting the atom’s spin. The solutions are expressed in the form of hyperbolic function solutions that have different physical meanings such that the hyperbolic tangent appears in the calculation and rapidity of special relativity while, the hyperbolic cotangent arises in the Langevin function for magnetic polarization, the hyperbolic secant arises in the profile of a laminar jet. The various kinds of soliton solutions in single and combined form like bright, dark, singular as well as bright-dark and singular in the mixed form are also extracted by the mean of extended sinh-Gordon equation expansion method. By using the appropriate values of the involved parameters, 3D, 2D and their corresponding contour graphs are sketched for physical movement of the attained results. We also discuss the modulation instability (MI) analysis of the governing model. The constraint conditions for the existence of soliton solutions are also mentioned. The calculated work and earned results show the power, effectiveness, and the simplicity of applied method to discuss the soliton solutions as the contrast with other analytical schemes. The main outcome of the proposed technique is that we have succeeded in a single move to get and organize various types of new solutions.

本文研究了三耦合Gross-Pitaevskii（GP）系统（也称为非线性Schrödinger（NLS）方程）的光学孤子，它描述了 $F=\mathrm{1个}$ 玻色子-玻色-爱因斯坦凝结， $F$表示原子的自旋。这些解以具有不同物理含义的双曲函数解的形式表示，使得双曲正切出现在狭义相对论的计算和快速度中，而双曲正切出现在朗格万函数的磁极化作用中，双曲正割出现在层流射流的轮廓。通过扩展sinh-Gordon方程展开法的平均值，还提取了单一形式和组合形式的各种孤子解，例如亮，暗，奇异以及亮暗和奇异的混合形式。通过使用所涉及参数的适当值，可以绘制3D，2D及其相应的轮廓图，以实现所获得结果的物理运动。我们还将讨论控制模型的调制不稳定性（MI）分析。还提到了孤子解存在的约束条件。计算的工作和所得的结果显示了讨论孤子解的方法的强大，有效和简便，这是与其他分析方案的对比。所提出的技术的主要结果是，我们成功地采取了一项行动，以获取和组织各种类型的新解决方案。