Uncovering New Wave Profiles in Boussinesq-Type KdV Systems Through Symbolic and Semi-Analytical Methods

We study here the Boussinesq-type Korteweg–de Vries (KdV) equation, a nonlinear partial differential equation, for describing the wave propagation of long, nonlinear, and dispersive waves in shallow water and other physical scenarios. In order to obtain novel families of wave solutions, we apply two efficient analytical techniques: the Modified Extended tanh (ME-tanh) method and the Modified Residual Power Series Method (mRPSM). These methods are used for the very first time in this equation to produce both exact and high-order approximate solutions with rich wave behaviors including soliton formation and energy localization. The ME-tanh method produces a rich class of closed-form soliton solutions via systematic simplification of the PDE into simple ordinary differential forms that are readily solved, while the mRPSM produces fast-convergent approximate solutions via a power series representation by iteration. The accuracy and validity of the results are validated using symbolic computation programs such as Maple and Mathematica. The study not only enriches the current solution set of the Boussinesq-type KdV equation but also demonstrates the efficiency of hybrid analytical techniques in uncovering sophisticated wave patterns in multimensional spaces. Our findings find application in coastal hydrodynamics, nonlinear optics, geophysics, and the theory of elasticity, where accurate modeling of wave evolution is significant.