Geometric formulation and multi-dark soliton solution to the defocusinig complex short pulse equation

TL;DR

This paper explores the geometric and algebraic structures of the defocusing complex short pulse equation, establishing links with space curve motions, deriving it from the KP hierarchy, and providing multi-dark soliton solutions.

Contribution

It introduces a geometric framework connecting the defocusing CSP equation with space curves and derives the equation from the extended KP hierarchy, also providing explicit multi-dark soliton solutions.

Findings

Established a geometric link between the CCD system and space curves in Minkowski space.

Derived the defocusing CSP equation from the extended KP hierarchy.

Constructed multi-dark soliton solutions in determinant form.

Abstract

In the present paper, we study the defocusing complex short pulse (CSP) equations both geometrically and algebraically. From the geometric point of view, we establish a link of the complex coupled dispersionless (CCD) system with the motion of space curves in Minkowski space $\mathbf{R}^{2,1}$, then with the defocusing CSP equation via a hodograph (reciprocal) transformation, the Lax pair is constructed naturally for the defocusing CSP equation. We also show that the CCD system of both the focusing and defocusing types can be derived from the fundamental forms of surfaces such that their curve flows are formulated. In the second part of the paper, we derive the the defocusing CSP equation from the single-component extended KP hierarchy by the reduction method. As a by-product, the $N$-dark soliton solution for the defocusing CSP equation in the form of determinants for these equations is provided.