We carry out the hidden structural symmetries embedded within a system comprising ultra-short pulses which propagate in optical nonlinear media. Based upon the Wahlquist-Estabrook approach, we construct the Liealgebra valued connections associated to the previous symmetries while deriving their corresponding Lax-pairs, which are particularly useful in soliton theory. In the wake of previous results, we extend the above prolongation scheme to higher-dimensional systems from which a new (2+ 1)-dimensional ultra-short pulse equation is unveiled along with its inverse scattering formulation, the application of which are straightforward in nonlinear optics where an additional propagating dimension deserves some attention.
We focus our attention on the coupled short-pulse equation recently derived by Feng [J. Phys. A: Math. Theor. 45, 085202 (2012)] from a two-dimensional Bäcklund transformation of the Toda lattice equation. Investigating the prolongation structure of such a system, we unveil the hidden structural symmetry that governs the dynamics of the wave solutions to the system alongside with the corresponding Lax-pairs. As a matter of illustration, following the Wadati-Konno-Ichikawa scheme, we construct some solitary wave solutions to the system and study their interactions.
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