New Exact Solutions and the Dynamical Behavior for the (2+1)-Dimensional Kundu-Mukherjee-Naskar Equation

This study presents novel exact solutions for the (2+1)-dimensional Kundu–Mukherjee–Naskar (KMN) equation, a model governing optical soliton propagation in nonlinear media. By incorporating two arbitrary functions G ( y ) and \( H(\xi ) \) into the solution ansatz. Four types of solutions are constructed: (i) basic solitons exhibiting cross-shaped breather dynamics; (ii) periodic breathers combining rational spatial localization with temporal oscillations; (iii) localized breathers featuring static periodic backgrounds coupled to moving bright solitons; and (iv) fractal rogue waves emerging from rational fractional functions. These solutions exhibit rich dynamics including spatiotemporal coupling dynamics, breathing behaviors, directional propagation ( \( v_x = \alpha k \) ), and extreme localization of wave energy. Parameter analysis demonstrates control over amplitude ( C ), localization( K , L , P ), and energy distribution \(( A_0, B_0 )\) , with numerical simulations visually confirming the predicted structures. These results significantly expand the KMN solution space, enhancing the modeling of complex wave phenomena in optical materials like birefringent fibers and photonic crystals.