Some Salient Features of a (3 + 1)-Dimensional Nonlinear Integro-Differential Evolution Equation: Lie Symmetries, Traveling Waves and Conservation Laws
Yanga Gaxela, Abdullahi Rashid Adem, Ben Muatjetjeja, Sivenathi Oscar Mbusi, Ahmed H. Arnous, Anjan Biswas
Source abstract
A generalized (3+1)-dimensional Hirota-type nonlinear wave equation is investigated by combining Lie point symmetry analysis, invariant reductions, exact traveling-wave construction, and the direct multiplier method. A potential formulation is introduced under an explicit decay condition that makes the nonlocal term well defined. The admitted point symmetries are re-derived from the prolonged invariance criterion, and several symmetry reductions are obtained, including mixed translations, a transverse shear, scaling, and a general traveling-wave phase. The reduced traveling-wave equation admits an affine Riccati representation with hyperbolic, trigonometric, rational, and Bernoulli-type branches under explicit parameter restrictions. The corresponding 3D, 2D, and density visualizations are retained to illustrate the resulting wave profiles. Zeroth-order multipliers are then used to construct local conserved vectors in divergence form. The analysis provides a verified symmetry and conservation-law framework for the proposed higher-dimensional Hirota-type extension and supplies exact solutions that can be used as analytical benchmarks. The transverse-variable reductions and the zeroth-order multiplier family are derived specifically for the present (3+1)-dimensional potential equation, while the Riccati functional forms are used as standard representations rather than claimed as new special functions.
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