Solving the coupled Gerdjikov-Ivanov equation via Riemann-Hilbert approach on the half line

Area of Science:

• Nonlinear Partial Differential Equations
• Mathematical Physics
• Soliton Theory

Background:

• The coupled Gerdjikov-Ivanov equation is a significant model in nonlinear science.
• Solving such equations often involves complex analytical and numerical methods.
• Understanding soliton solutions is crucial for applications in various physical phenomena.

Purpose of the Study:

• To investigate the coupled Gerdjikov-Ivanov equation on a half-line interval.
• To apply the Fokas method and Riemann-Hilbert technique for finding solutions.
• To derive a general pattern for N-soliton solutions.

Main Methods:

• Utilizing the Fokas method for spectral analysis.
• Employing the Riemann-Hilbert technique to construct the potential function.
• Analyzing spectral functions and their interdependencies via a compatibility condition.
• Solving associated regular and non-regular Riemann-Hilbert problems.

Main Results:

• Established a global connection and compatibility condition for spectral functions.
• Successfully converted the initial value problem into a Riemann-Hilbert problem.
• Derived a general pattern for N-soliton solutions of the coupled Gerdjikov-Ivanov equation.

Conclusions:

• The Fokas method and Riemann-Hilbert technique are effective for solving the coupled Gerdjikov-Ivanov equation.
• Spectral functions are correlated, not independent, obeying a global connection.
• The study provides a framework for understanding complex soliton dynamics.