Ring-like double-breathers in the partially nonlocal medium with different diffraction characteristics in both directions under the external potential

Using a converting relation, a (3+1)-dimensional partially nonlocal nonlinear Schrödinger model with different diffraction characteristics in both directions under the external potential is reduced into a (2+1)-dimensional model, and ring-like double-breather solution is reported. Partial nonlocal properties and evolution of ring-like double-breathers are respectively revealed in the exponential system with the constant and exponential chirp phases for the constant linear phase. In both cases, for the same thickness parameter W, the radius parameter R determines the radius of ring and cylinder structures of the ring-like double-breathers. For the same radius parameter R, the thickness parameter W decides the spacing between two rings and between the hollow cylinders of the ring-like double-breathers. With the adding Hermite parameter p, along the z -axis, the layer number of ring-like double-breathers increase as p+1.