<p>In order to gain a deeper understanding of nonlinear wave dynamics, soliton solutions play an important role. In this research, using suitable transformations, we rigorously examine a wide range of wave solutions for the nonlinear Murray equation, exhibiting new wave patterns, including lump, lump with one kink, lump with two kink, rogue waves, periodic, breathers and multi-waves solutions. In individuals with cardiovascular illness, these solutions can be utilized to increase blood flow since they reflect a consistent fluctuation in blood vessel shape and diameter. Under the pathological conditions, localized pressure pulse occur in arteries, for which lump type solutions can be used as mathematical analogues. Rogue waves can describe sudden extreme spikes in blood pressure. Moreover, multi-wave solutions play in important role in hemodynamics, particularly in modeling the nonlinear arterial systems. These solutions have recently emerged in the literature, providing researchers with a crucial tool for understanding intricate biological systems. To illustrate the characteristics and development of the solutions, comprehensive graphical representations are provided, including 3D surface plots, 2D profiles, and contour plots. By examining these plots, we analyze how parameters affect the amplitude and propagation behavior of nonlinear waves which are related to change in the geometry of blood vessels. The derived solutions can deepen our comprehension of complex biological systems and may help in the development of new medical models and treatments. To the best of our knowledge, this is the first study to report such a diverse class of exact wave solutions for the nonlinear Murray equation with heterogeneous wall properties.</p>