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<p>This study focuses on the vector Lakshmanan–Porsezian–Daniel (LPD) equation, which describes the propagation of ultrashort pulses in birefringent or two-mode fibers. We solve the vector LPD equation under the condition of zero boundary conditions at infinity using the inverse scattering method. First, the corresponding Jost solutions are constructed, and their analytical, asymptotic, and symmetry properties are discussed–along with those of the scattering coefficients. Next, the Riemann–Hilbert problem is formulated by modifying the Jost solutions; by directly eliminating the poles of the reflection coefficients, this problem is further transformed into a regular one. Based on the asymptotic properties of the Riemann–Hilbert problem, we derive the formulas for the Nth-order soliton solutions of the vector LPD equation. Using these formulas, explicit expressions for the first- and second-order solitons are presented, along with their characteristic patterns. For the second-order soliton: when it has two distinct velocities, energy transfer occurs between its components before and after collision (while the total energy remains conserved); when it has two identical velocities, it exhibits a bound state, which can form a soliton molecule by selecting specific parameters. This work is expected to provide a theoretical basis for understanding the physical mechanism of energy transfer from the pump to the signal.</p>

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Riemann–Hilbert method for the vector Lakshmanan–Porsezian–Daniel equation and energy-sharing collisions of solitons in optical fiber communications

  • 期刊论文

  • 发表日期：

    2026年3月10日

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  • Jie Zheng,
  • Yafang Zhao,
  • Yongshuai Zhang,
  • Fangyu Tian,
  • Wei Liu

摘要

This study focuses on the vector Lakshmanan–Porsezian–Daniel (LPD) equation, which describes the propagation of ultrashort pulses in birefringent or two-mode fibers. We solve the vector LPD equation under the condition of zero boundary conditions at infinity using the inverse scattering method. First, the corresponding Jost solutions are constructed, and their analytical, asymptotic, and symmetry properties are discussed–along with those of the scattering coefficients. Next, the Riemann–Hilbert problem is formulated by modifying the Jost solutions; by directly eliminating the poles of the reflection coefficients, this problem is further transformed into a regular one. Based on the asymptotic properties of the Riemann–Hilbert problem, we derive the formulas for the Nth-order soliton solutions of the vector LPD equation. Using these formulas, explicit expressions for the first- and second-order solitons are presented, along with their characteristic patterns. For the second-order soliton: when it has two distinct velocities, energy transfer occurs between its components before and after collision (while the total energy remains conserved); when it has two identical velocities, it exhibits a bound state, which can form a soliton molecule by selecting specific parameters. This work is expected to provide a theoretical basis for understanding the physical mechanism of energy transfer from the pump to the signal.