The generalized q-deformed sinh Gordon equation (GDSGE) serves as a significant nonlinear partial differential equation with profound applications in physics. This study investigates the GDSGE's mathematical and physical properties, examining its solutions and clarifying the essence of the q-deformation parameter. The Sardar sub-equation method (SSEM) and sine-Gordon expansion method (SGEM) are employed to solve this GDSGE. The synergistic application of these techniques improves our knowledge of the GDSGE and provides a thorough foundation for investigating different evolution models arising in various branches of mathematics and physics. A positive aspect of the proposed methods is that they offer a wide variety of solitons, including bright, singular, dark, combination dark-singular, combined dark-bright, and periodic singular solitons. Obtained solutions demonstrate the method's high degree of reliability, simplicity, and functionalization for various nonlinear equations. To better describe the physical characterization of solutions, a few 2D and 3D visualizations are generated by taking precise values for parameters using mathematical software, Mathematica.

New traveling wave solutions for the nonlinear fractional Schrödinger equation (FSE), obtained using conformable fractional derivatives, are presented in this paper. Despite extensive research on classical and fractional Schrödinger models, a systematic development of accurate traveling wave solutions employing conformable operators in conjunction with effective symbolic approaches remains lacking. To bridge this gap, we employ a Hamiltonian-based technique, a variational formulation via the Ritz method, and the modified Sardar subequation method. The fractional governing model is reduced to a nonlinear ordinary differential equation through a complex traveling wave transformation, which is analytically solved to yield new families of solutions. Two- and three-dimensional graphical representations of the solution's physical properties are presented, emphasizing the wave dynamics of the proposed fractional model.