Defocusing complex short-pulse equation and its multi-dark-soliton solution

Abstract

In this paper, we propose a complex short pulse equation of both focusing and defocusing types, which governs the propagation of ultra-short pulses in nonlinear optical fibers. It can be viewed as an analogue of the nonlinear Schrödinger (NLS) equation in the ultra-short pulse regime. Furthermore, we construct the multi-dark soliton solution for the defocusing complex short pulse equation through the Darboux transformation and reciprocal (hodograph) transformation. Oneand two-dark soliton solutions are given explicitly, whose properties and dynamics are analyzed and illustrated.

Multi-soliton, multi-breather and higher order rogue wave solutions to the complex short pulse equation

“…The complex short pulse equation of both focusing and defocusing types can be derived from the context of nonlinear optics. The results are summarized in a separate work [50]. Proof: Fixed y − v k s = const, and s → −∞, it follows that…”

Geometric Formulation and Multi‐dark Soliton Solution to the Defocusing Complex Short Pulse Equation

“…In [16], a defocusing CSP equation was derived from physical context in nonlinear optics as an analog of the NLS equation in ultra-SP regime.…”

“…from Eq. (16). Thus, the link of the defocusing CCD system to the motion of space curves in Minkowski space R 2,1 is established.…”

“…In this case, the envelope of the dark soliton is smooth. However, as analyzed in [16], if we take β = (2 + 2 √ 7)/3 while keeping other parameters unchanged, the dark soliton becomes a cusped envelope soliton, as shown in Figure 3. In other words, the dark solution has more tendency to become singular (cusped or looped one) as the amplitude of the background waves increases.…”

“…It turns out that this is indeed the case. Same as the focusing CSP equation discussed in [1,5], the defocusing CSP equation can also occur in nonlinear optics when ultra short pulses propagate in a nonlinear media of defocusing type [16].…”