<p>Investigating different types of solitons in heat transport presents promising opportunities for information processing technologies. In this paper, the heat conduction equation is analyzed using Lie symmetry analysis. We compute the optimal system of one-dimensional subalgebras for the heat conduction equation and utilize it to identify a set of invariant solutions. Additionally, we employ the Lie group method to reduce the heat conduction equation to new differential equations. Furthermore, we apply the generalized Riccati equation mapping method to extract solutions in the forms of kink, singular, dark-singular, and bright-kink solitons. We present our findings through 3D, density, 2D, and contour plots. In this paper, we extend our investigation to include modulation instability and gain spectrum analysis within the framework of nonlinear dynamical systems. Although the idea of using heat solitons for information transfer is still largely theoretical, it opens up an exciting area for further exploration. Studying these solitons through the lens of information theory provides new motivation for research, emphasizing their potential contribution to the development of heat-driven information systems. For novelty, we conducted a detailed comparison of our solutions with existing methods, highlighting the advantages and unique features of our approach.</p>