In this article, a fully discrete short pulse (SP) equation is presented as an integrability condition of a linear system of difference equations (also known as discrete Lax pair). Additionally, two semi-discrete versions of the SP equation have also been obtained from fully discrete SP equation under continuum limits. Darboux transformation is employed to compute multi-soliton solutions of fully discrete and semi-discrete SP equations. Explicit expressions of first and second nontrivial soliton solutions are computed. We also derived explicit expression of breather solution for fully discrete SP equation. The dynamics of single loop soliton and interaction mechanism of loop-loop and loop-antiloop solutions has been explored and illustrated.

翻译：在本条中,完全离散短脉冲(SP)方程式作为差异方程式线性系统(又称离散拉克斯对方)的融合性条件提出,此外,还从连续限制下完全离散的SP方程式中获得了两个半分解的SP方程式版本,Darboux转换用于计算完全离散和半分解的SP方程式的多索尔顿解决方案,计算出第一和第二个非三角索利顿方程式的清晰表达方式。我们还为完全离散的SP方程式生成了呼吸器解决方案的清晰表达方式。探索并演示了单环状索利通的动态和环状和环状内孔解决方案的互动机制。