Optimal Classifications, New Doubly Periodic, Multiple Soliton Solutions, Waves Dynamics and Conserved Quantities of a New Integrable (1+1)-D Boussinesq Equation with Dissipative Phase in Fluid Dynamics - Adeyemo, Oke Davies (Journal of Applied Nonlinear Dynamics, 2026)
The theory of natural evolution and computations has contributed to many advancements in science and engineering. Consequently, this article explores the analytical examinations of a new integrable (1+1)-dimensional Boussinesq equation recently formulated in the literature which is applicable in optics, fluid dynamics, ocean science and other nonlinear sciences and engineering. Lie group theory in differential equations is then applied to identify point symmetries within the model, enabling the derivation of nonlinear ordinary differential equations through symmetry reductions. In addition, direct integration is invoked to secure some soliton solutions, such as bright, periodic, and singular. More importantly, the Jacobi elliptic function approach is further engaged to secure abundant general exact travelling wave solutions in the structures of periodic as well as singular soliton solutions to the model. This technique enables the attainment of various exact soliton solutions, including topological and non-topological soliton solutions (both complex and non-complex). Additionally, general periodic function solutions of note, such as cosine amplitude, sine amplitude, and delta amplitude solutions of the model, are also secured. In the same vein, a power series approach is utilized to solve part of the difficult nonlinear ordinary differential equations obtained. Besides, by taking some essential limits of part of the solutions, one obtains various soliton results, including triangular solutions of the understudy model. These are in the form of hyperbolic and trigonometric functions, achieved with regards to the secant, tangent, and cotangent functions. Furthermore, numerical simulations of the solutions are invoked to gain a gross knowledge of the physical phenomena represented by the understudy integrable Boussinesq equation. Conclusively, the study further produces conserved quantities of note, such as energy, mass, and momentum, which are secured through the use of Ibragimov’s theorem, as well as the multiplier approach.