Exact Travelling-Wave Solutions of a Nonlinear Convection–Diffusion Equation with Square-Root Flux

We investigate a nonlinear convection—diffusion equation involving a non-polynomial square-root flux. By applying a travelling-wave reduction and introducing a structurally motivated square-root transformation, we show that the resulting ordinary differential equation possesses an intrinsic logistic first-order structure. This reduction yields an explicit closed-form monotone travelling-wave solution connecting two equilibrium states and uniquely determines the admissible wave speed c=2/3. The solution describes the diffusive smoothing of an initial discontinuity into a propagating transition layer. Direct finite-difference simulations with Riemann-type initial data confirm convergence toward the analytical profile and verify the predicted wave speed. These results demonstrate that convection—diffusion equations with non-algebraic flux functions can admit exact travelling-wave solutions when appropriate structural transformations are identified, providing both theoretical insight and reliable benchmarks for numerical methods.