This paper studies the self-similar solutions of a generalized Burgers equation with nonlinear damping. The authors analyze the initial value problem for the resulting ordinary differential equation and observe a wide variety of solutions, including positive single humps, unbounded solutions, and those with finite zeros. They prove the existence and non-existence of positive bounded solutions with different types of decay (exponential or algebraic) to zero at infinity for specific parameter ranges. The analysis reveals a rich structure of solutions, including the existence of solutions with exponential decay to zero as $\eta \to \infty$ and algebraic decay to zero as $\eta \to -\infty$, as well as solutions with finite zeros on either side of the real line. The paper also includes numerical studies and detailed proofs of various theorems characterizing the behavior of these solutions.This paper studies the self-similar solutions of a generalized Burgers equation with nonlinear damping. The authors analyze the initial value problem for the resulting ordinary differential equation and observe a wide variety of solutions, including positive single humps, unbounded solutions, and those with finite zeros. They prove the existence and non-existence of positive bounded solutions with different types of decay (exponential or algebraic) to zero at infinity for specific parameter ranges. The analysis reveals a rich structure of solutions, including the existence of solutions with exponential decay to zero as $\eta \to \infty$ and algebraic decay to zero as $\eta \to -\infty$, as well as solutions with finite zeros on either side of the real line. The paper also includes numerical studies and detailed proofs of various theorems characterizing the behavior of these solutions.