Abstract
In this article we obtain generalized semi-discrete short pulse equation through discretization of the associated linear eigenvalue problem of the short pulse equation. Through implication of different symmetry reductions on the generalized semi-discrete short pulse equation, new integrable equations are assembled such as complex and the PT-symmetric nonlocal semi-discrete short pulse equation. Darboux transformation is applied to construct multi-soliton solutions and we obtain a generalized formula to compute higher-order nontrivial solutions of the generalized semi-discrete short pulse equation. To demonstrate the dynamics of solutions, explicit expressions of nontrivial solutions up-to second-order are investigated for the PT-symmetric nonlocal semi-discrete short pulse equation. It is interesting to add that at every stage the solutions obtained in this work are reduced to already obtained results of reverse space–time PT-symmetric short pulse equation under continuum limit. Interactions between loops of attractive and repulsive types and first-order breather solution under local and nonlocal reductions are illustrated first time in this article for local and PT-symmetric nonlocal semi-discrete short pulse equation.
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