Darboux transformation and positons of the inhomogeneous Hirota and the Maxwell-Bloch equation

In this paper, we derive Darboux transformation of the inhomogeneous Hirota and the Maxwell-Bloch(IH-MB) equations which is governed by femtosecond pulse propagation through inhomogeneous doped fibre. The determinant representation of Darboux transformation is used to derive soliton solutions, positon solutions of the IH-MB equations.

💡 Research Summary

This paper addresses the integrable inhomogeneous Hirota–Maxwell‑Bloch (IH‑MB) system, which models femtosecond pulse propagation in an inhomogeneously doped optical fiber. The authors first formulate the Lax pair for the IH‑MB equations, introducing space‑dependent coefficients a₁(z)…a₇(z) and b₁(z), b₂(z) that represent group‑velocity dispersion, third‑order dispersion, self‑steepening, self‑phase modulation, gain/loss, and the coupling between the optical field and the two‑level resonant medium. Consistency conditions a₂=∂ₙb₁/∂z, a₅=6a₄b₂⁻¹, a₆=2a₃b₂⁻¹ are imposed to preserve the zero‑curvature condition and guarantee complete integrability.

Using the AKNS framework, the authors construct a Darboux transformation (DT) of the form T(λ)=λI−S, where S is built from eigenfunctions of the Lax pair evaluated at two distinct spectral parameters λ₁ and λ₂. By enforcing the reduction λ₂=−λ₁* and the Hermitian symmetry s₂₁=s₁₂*, the transformed fields remain physically admissible (real population inversion, complex‑conjugate polarization). The DT satisfies the intertwining relations Tₜ+TU=U⁽¹⁾T and T_z+TV=V⁽¹⁾T, yielding explicit formulas for the new electric field E⁽¹⁾, polarization p⁽¹⁾, and inversion η⁽¹⁾. In particular, E⁽¹⁾=E+2b₁⁻¹s₁₂ and the transformed MB matrix V⁽¹⁾{‑1}=(S+iωb₂)V{‑1}(S+iωb₂)⁻¹.

A major contribution of the work is the determinant representation of the DT. For the one‑fold case, the authors define a 2×2 matrix T₁(λ) whose entries are expressed as ratios of 2×2 determinants built from the eigenfunctions Φ₁, Φ₂. The denominator Δ₁ is the Wronskian‑type determinant of the two eigenfunctions. For the two‑fold case, a 2×2 matrix T₂(λ) is constructed using four eigenfunctions, with a 4×4 determinant Δ₂ in the denominator. These determinant formulas allow systematic computation of higher‑order DTs and ensure that T(λ) annihilates the corresponding eigenvectors at λ=λ_i, i.e., T(λ_i)