In this study, we examined a (2+1)-dimensional generalized breaking soliton system (GBSS) using both analytical and numerical methods. By applying a generalized direct algebraic method, we derived exact solutions that displayed a variety of solitary and periodic wave patterns. These solutions illuminated the interplay between nonlinearity and dispersion in several physical contexts, including fluid dynamics, plasma physics, and nonlinear optics. In addition, we developed a robust numerical scheme employing an adaptive moving mesh technique based on the MMPDE5 framework. Stability and error analyses confirmed that this method concentrated grid points around steep gradients, achieved second-order spatial convergence, and enhanced computational efficiency. By comparing numerical and exact solutions, we provided more profound insights into GBSS dynamics and facilitated future investigations of complex, multidimensional, and nonlinear wave phenomena.

In this study, we investigate the traveling wave solutions of the Gilson-Pickering equation using two different approaches: F-expansion and (1/G ${}^{\mathrm{\prime }}$)-expansion. To carry out the analysis, we perform a numerical study using the implicit finite difference approach on a uniform mesh and the parabolic-Monge-Ampère (PMA) method on a moving mesh. We examine the truncation error, stability, and convergence of the difference scheme implemented on a fixed mesh. MATLAB software generates accurate representations of the solution based on specified parameter values by creating 3D and 2D graphs. Numerical simulations with the finite difference scheme demonstrate excellent agreement with the analytical solutions, further confirming the validity of our approaches. Convergence analysis confirms the stability and high accuracy of the implemented scheme. Notably, the PMA method performs better in capturing intricate wave interactions and dynamics that are not readily achievable with a fixed mesh.

This paper presents an adaptive computational framework for the two-dimensional regularized long-wave (RLW) equation. The main focus was on the numerical treatment of the model on moving nonuniform meshes. The RLW equation was first rewritten in a mixed differential-algebraic (DA) form by introducing an auxiliary variable. This reformulation avoids the direct discretization of the mixed space–time derivatives and leads to a more convenient adaptive implementation. The spatial approximation was constructed on bidirectionally adaptive tensor-product grids using five-point Fornberg finite-difference operators. Mesh redistribution was performed through a parabolic Monge–Ampère (PMA) rezoning strategy combined with monitor functions and shape-preserving interpolation between successive meshes. Exact travelling-wave solutions were also derived through an oblique wave reduction. These solutions were used mainly as benchmark data for the numerical computations. In particular, the solitary-wave solution provides compatible initial conditions, boundary data, auxiliary-field values, and reference profiles for error measurement. The stability discussion was carried out under admissible mesh evolution. On each frozen mesh, the spatial approximation was formally second-order consistent. The convergence behavior of the full adaptive algorithm was assessed numerically because mesh redistribution and inter-mesh interpolation introduce additional error effects. Numerical experiments were presented for single-wave and two-pulse configurations. The results show that the adaptive method preserves the wave structure with good accuracy and produces smaller errors than the corresponding fixed-mesh computation. The adaptive meshes remained concentrated near the dominant wave regions while maintaining global mesh regularity throughout the simulation.