A novel multi-component generalization of the short pulse equation and its multisoliton solutions

TL;DR

This paper introduces a new multi-component generalization of the short pulse equation, providing explicit multisoliton solutions expressed via pfaffians and demonstrating the system's complete integrability.

Contribution

It presents a novel multi-component system extending the short pulse equation, with explicit multisoliton solutions and a proof of integrability via a Lax pair.

Findings

Solutions expressed in pfaffians differ from determinantal solutions.

The 2-component system is shown to be completely integrable.

Detailed analysis of solitonic properties like loops and breathers.

Abstract

We propose a novel multi-component system of nonlinear equations that generalizes the short pulse (SP) equation describing the propagation of ultra-short pulses in optical fibers. By means of the bilinear formalism combined with a hodograph transformation, we obtain its multi-soliton solutions in the form of a parametric representation. Notably, unlike the determinantal solutions of the SP equation, the proposed system is found to exhibit solutions expressed in terms of pfaffians. The proof of the solutions is performed within the framework of an elementary theory of determinants. The reduced 2-component system deserves a special consideration. In particular, we show by establishing a Lax pair that the system is completely integrable. The properties of solutions such as loop solitons and breathers are investigated in detail, confirming their solitonic behavior. A variant of the 2-component system is also discussed with its multisoliton solutions.