Dynamics of nonlinear bright waves and multi-bright wave for the (2+1)-dimensional Caudrey-Dodd-Gibbon-Kotera-Sawada equation

The (2+1)-dimensional Caudrey–Dodd–Gibbon–Kotera–Sawada (CDGKS) equation, a classical integrable model governing nonlinear wave propagation in shallow water, plasma physics, and nonlinear optics, is investigated analytically using two complementary methods: Hirota’s bilinear method and the long-wave limit technique. Applying Hirota’s method, we derive the complete multi-soliton hierarchy from \(N=1\) to \(N=5\) , with explicit \(\tau \) -function representations and exact phase-shift coefficients \(A_{ij}\) . Via the long-wave limit, we construct one-, two-, and three-lump solutions as rational, algebraically localised wave structures with analytic propagation trajectories. Imposing complex-conjugate constraints on the two-soliton \(\tau \) -function yields breather solutions with spatially localised, temporally periodic profiles. Mixed interaction solutions are constructed by selectively combining lump, breather, and soliton components, producing lump–breather, lump–soliton, and lump–two-soliton configurations. Finally, resonance conditions ( \(A_{ij}=0\) ) are derived that generate Y-shaped and X-shaped fission/fusion wave patterns. The results are illustrated through 3D surface and 2D contour plots, with explicit parameter dependence of wave amplitude, width, propagation velocity, phase shift, and oscillation frequency reported analytically. These findings advance the understanding of integrable nonlinear systems and have direct relevance to wave energy transport in fluid dynamics, modulated wave packets in plasma, and spatially localised pulses in nonlinear optics.