Investigating exact solutions for the (3+1)-dimensional KdV-CBS equation: A non-traveling wave approach

Non-traveling wave solutions are crucial, as they provide deeper insights into the complex dynamics and diverse wave structures of nonlinear systems, expanding the understanding of phenomena beyond traditional traveling wave approaches. This research focuses on deriving explicit non-traveling wave solutions for the (3+1)-dimensional KdV–Calogero–Bogoyavlenskii–Schiff (KdV-CBS) equation. A new method using an improved variable separation technique is applied to find abundant explicit non-traveling wave solutions. This technique integrates elements from both the KdV and CBS equations, extending and unifying previous methodologies. The derived solutions incorporate multiple arbitrary functions, showcasing greater versatility than previous methodologies. By selecting specific forms for these functions, diverse non-traveling exact solutions such as periodic solitary waves and cross soliton-like patterns are constructed. All derived solutions are validated by plugging them into the original equation using Maple software, confirming their correctness. Since non-traveling wave solutions for the (3+1)-dimensional KdV-CBS equation have not been thoroughly explored, this study makes a significant contribution to the field.