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

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.

Minor corrections

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• Lax Representations for Matrix Short Pulse Equations
• Parametric solutions of the generalized short pulse equations
• Soliton interactions and Yang-Baxter maps for the complex coupled short-pulse equation
• Super Extensions of the Short Pulse Equation
• Discontinuous Galerkin methods for short pulse type equations via hodograph transformations
• Some Wadati-Konno-Ichikawa type integrable systems and their constructions
• Integrable negative flows of the Heisenberg ferromagnet equation hierarchy
• Liouville correspondence between the Short-Pulse Hierarchy and the Sine-Gordon Hierarchy