Optimal System, Reductions, and Solutions for One-Dimensional Generalized Korteweg–de Vries Equation

In this paper, we investigate a generalized form of the Korteweg–de Vries (gKdV) equation, which serves as an important mathematical model for simulating the propagation of long waves, including tsunami waves in shallow water environments. The generalized KdV framework provides a more flexible and comprehensive description of wave dynamics, allowing for the analysis of the nonlinear effects that arise in realistic physical settings. To analyze this model, we employ a combination of analytical techniques, including the symmetry (Lie group) method, the direct integration approach, the unified Riccati method and the explicit power series method. These methods allow us to derive a variety of exact, closed-form solutions. In addition to obtaining exact solutions, we examine the conservation properties of the generalized KdV equation. Using both the multiplier method and classical Noether theorem, we derive several conservation laws associated with the model. Among these are the fundamental conservation of momentum and conservation of energy, which play a crucial role in understanding the physical consistency and stability of the system. Overall, this work highlights both the analytical richness of the generalized KdV equation and its relevance in modeling nonlinear wave phenomena such as tsunami propagation.