The loop rogue wave solutions for the Wadati-Konno-Ichikawa equation

Area of Science:

• Nonlinear physics
• Mathematical physics
• Wave phenomena

Background:

• The Wadati-Konno-Ichikawa equation models complex wave phenomena.
• Rogue waves are extreme amplitude events with significant implications.
• Understanding rogue wave formation is crucial in various scientific fields.

Purpose of the Study:

• To derive and analyze the first-order rogue wave solution of the Wadati-Konno-Ichikawa equation.
• To investigate the role of arbitrary parameters, particularly background wave amplitude, on the solution's properties.
• To characterize the formation and features of singular rogue wave solutions, including loop-type profiles.

Main Methods:

• Application of the Darboux transformation to generate solutions.
• Utilizing the inverse hodograph transformation for solution analysis.
• Detailed examination of the analyticity conditions of the rogue wave solution.

Main Results:

• A first-order rogue wave solution with two arbitrary parameters was successfully generated.
• The analyticity of the rogue wave solution is critically dependent on the background wave amplitude parameter.
• Singular solutions exhibit a distinct loop-type profile, with their general characteristics elucidated.

Conclusions:

• The background wave amplitude is a key factor in determining the nature of rogue wave solutions.
• The study provides a detailed description of loop rogue waves, offering new insights into extreme wave events.
• The findings contribute to the theoretical understanding of nonlinear wave dynamics and rogue wave phenomena.