Abundant analytical solutions for width-Burger equation: M-shaped, multiwave, breathers and their interactions

In order to better understand nonlinear wave dynamics, we systematically investigate a wide variety of exact solutions for the modified equal width-Burgers equation (mEWBE), revealing new wave patterns. Lump (L), lump with one kink $(L_{1}K)$ , lump with two kinks $(L_{2}K)$ , multiwave (MW), rogue wave (RW), periodic wave (PW), periodic cross-lump wave (PCLW), periodic cross-kink wave (PCKW), interaction between lump, periodic, and kink waves (LPKW), and breather lump wave (BLW) are among the systematic solutions that we obtain by using the appropriate transformation method. The M-shaped rational solution (MSRS), M-shaped rational solution with one kink $(MSR_{1}K)$ , M-shaped rational solution with two kinks $(MSR_{2}K)$ , periodic cross-rational solution (PCRS), kink-cross rational solution (KCRS), M-shaped rational solution with periodic and kink components (MSRPK), and M-shaped rational solution with rogue and kink features (MSRRK) are among the other novel rational solutions that we thoroughly examine. These results offer novel perspectives on intricate wave interactions in nonlinear systems. We further provide a deeper understanding of the propagation and interaction of effective wave structures in the WBE framework by presenting graphical visuals in many dimensions that highlight the complex characteristics of the found solutions. These results can be used practically to comprehend complicated wave behaviors in real-world systems, including shock waves, ocean waves, and multi-wave systems in engineering.