Exact traveling wave and soliton solutions for chemotaxis model and (3+1)-dimensional Boiti–Leon–Manna–Pempinelli equation

Abstract This study investigates exact traveling wave solutions for two important nonlinear models: the hyperbolic chemotaxis model and the (3+1)-dimensional Boiti–Leon–Manna–Pempinelli equation. Using the <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mfrac> <m:mrow> <m:msup> <m:mrow> <m:mi>G</m:mi> </m:mrow> <m:mrow> <m:mo accent="true">′</m:mo> </m:mrow> </m:msup> </m:mrow> <m:mrow> <m:msup> <m:mrow> <m:mi>G</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:msup> </m:mrow> </m:mfrac> </m:math> \frac{{G}^{^{\prime} }}{{G}^{2}} -expansion method, we derive solutions in the form of trigonometric, exponential, and rational functions, showcasing diverse wave behaviors such as periodic solitons, kink-type waves, and singular solitons. By visualizing these solutions with 3D and contour plots, we provide deeper insights into the spatial–temporal evolution of the wave dynamics. The findings highlight the versatility of the <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mfrac> <m:mrow> <m:msup> <m:mrow> <m:mi>G</m:mi> </m:mrow> <m:mrow> <m:mo accent="true">′</m:mo> </m:mrow> </m:msup> </m:mrow> <m:mrow> <m:msup> <m:mrow> <m:mi>G</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:msup> </m:mrow> </m:mfrac> </m:math> \frac{{G}^{^{\prime} }}{{G}^{2}} -expansion method for solving complex nonlinear equations and demonstrate its relevance in both biological systems, like chemotaxis, and physical wave dynamics.