Spatial structure of the non-integrable discrete defocusing Hirota equation

Area of Science:

• Nonlinear Dynamics
• Mathematical Physics
• Discrete Systems

Background:

• The non-integrable discrete defocusing Hirota equation is a significant model in nonlinear science.
• Understanding its spatial properties is crucial for applications in various physical phenomena.
• Previous studies have explored related discrete nonlinear equations, but a comprehensive analysis of the Hirota equation's spatial dynamics was lacking.

Purpose of the Study:

• To investigate the spatial properties of the non-integrable discrete defocusing Hirota equation.
• To construct and analyze periodic and aperiodic orbit solutions.
• To compare the spatial characteristics with the non-integrable discrete defocusing nonlinear Schrödinger equation.

Main Methods:

• Utilizing a planar nonlinear discrete dynamical map method.
• Constructing periodic orbit solutions for the stationary equation.
• Analyzing orbit behavior near special periodic solutions using the residue method.
• Employing numerical simulations to characterize parameter effects on aperiodic orbits.

Main Results:

• Periodic orbit solutions for the stationary discrete defocusing Hirota equation were successfully constructed.
• The analysis revealed that the non-integrable discrete defocusing Hirota equation exhibits more abundant spatial properties compared to the nonlinear Schrödinger equation.
• Numerical simulations demonstrated the influence of parameters on aperiodic orbits.
• A novel finding is the existence of triperiodic solutions for a reduced map, irrespective of the initial value.

Conclusions:

• The non-integrable discrete defocusing Hirota equation possesses complex and rich spatial properties.
• The planar nonlinear discrete dynamical map method is effective for analyzing such systems.
• The discovery of triperiodic solutions highlights new avenues for research in discrete nonlinear dynamics.