Dynamics of loop soliton solutions of $${\pmb {\mathcal {PT}}}$$-symmetric nonlocal short pulse equation

Abstract

In this article, we present a system of coupled short pulse equations as integrability condition of associated generalized linear eigenvalue problem. We obtain system of $$\mathcal {PT}$$-symmetric short pulse equation (SPE) by using suitable reduction condition. In order to investigate the dynamics of loop solitons for nonlocal SPE, we use hodograph transformation to transform standard SPE into an equivalent equation of motion and associated linear system. We construct multiple-soliton solutions for $$\mathcal {PT}$$-symmetric nonlocal SPE by employing Darboux transformation on the matrix-valued solution to the transformed linear system. We obtain a generalized formula for multiple solutions in terms of ratio of determinants. Families of exact loop and breather soliton solutions are computed by using the generalized formula. We analyze the impact of nonlocality on loops, breathers and interacting solitons for incoherent spectral parameters. We also characterize the conduct of the loop soliton interactions. Nonlocality-induced instabilities for loop and breather solutions are also illustrated. We also recover multiple loop soliton solutions of the classical SPE under local symmetry reduction.