Nonlinear soliton dynamics in the (2+1)-dimensional complex mKdV system in fluid dynamics

The research focuses on the (2+1)-dimensional complex modified Korteweg–de Vries system, which gives a better understanding of wave behavior through the capture of water particle’s nontrivial dynamics. The complex modified Korteweg–de Vries system is an important mathematical model in nonlinear sciences with potential applications in different areas, such as fluid dynamics, nonlinear optics, image processing and plasma physics. To investigate this model, we utilize a modified Sardar sub-equation method to obtain a wide variety of soliton solutions. These are periodic-singular solitons, bright solitons, singular solitons, dark solitons, combinations of bright–dark solitons, exponential solitons and rational solutions. The undetermined coefficients method is also utilized to obtain bright, dark and singular soliton solutions. In addition, we use the differential transform method to develop numerical solutions for the soliton solutions obtained by using the modified Sardar sub-equation method and the undetermined coefficients method. Our results help to expand the general knowledge of nonlinear wave structures, providing significant insights into mathematical physics and engineering applications.