Abstract A class of self-similar beams, named three-dimensional (3D) spatiotemporal parabolic accessible solitons, are introduced in the 3D highly nonlocal nonlinear media. We obtain exact solutions of the 3D spatiotemporal linear Schrodinger equation in parabolic cylindrical coordinates by using the method of separation of variables. The 3D localized structures are constructed with the help of the confluent hypergeometric Tricomi functions and the Hermite polynomials. Based on such an exact solution, we graphically display three different types of 3D beams: the Gaussian solitons, the ring necklace solitons, and the parabolic solitons, by choosing different mode parameters. We also perform direct numerical simulation to discuss the stability of local solutions. The procedure we follow provides a new method for the manipulation of spatiotemporal solitons.

中文翻译：

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三维抛物线柱坐标中的可访问孤子

摘要 在 3D 高度非局部非线性介质中引入了一类自相似光束，称为三维 (3D) 时空抛物线可访问孤子。我们通过变量分离的方法获得了抛物线柱坐标下3D时空线性薛定谔方程的精确解。3D 局部结构是在汇合超几何 Tricomi 函数和 Hermite 多项式的帮助下构建的。基于这样的精确解，我们通过选择不同的模式参数以图形方式显示三种不同类型的 3D 光束：高斯孤子、环形项链孤子和抛物线孤子。我们还执行直接数值模拟来讨论局部解的稳定性。