Dynamical System Structure of A Modified Equation

We study the dynamical–system structure underlying the viscous conservation law u_t+√u u_x=u_xx which arises in models of nonlinear transport with diffusion. By introducing a traveling–wave ansatz, the equation is reduced to a planar autonomous system whose phase–plane geometry completely characterizes steady wave profiles. The reduced system exhibits a continuous line of equilibria corresponding to uniform states, with stability determined by the relative magnitude of the wave speed and the characteristic speed √u. A critical manifold separates stable and unstable regimes and plays a role analogous to a sonic boundary. We demonstrate through phase-plane analysis that heteroclinic relations between usually unstable and normally stable equilibria correlate to viscous shock profiles, whereas rarefaction behavior results from the absence of such connections. The geometric framework makes the role of degeneracy on the nonlinear flux near vanishing states clear and offers a transparent interpretation of entropy admissibility. Both the general characteristics of scalar viscous conservation laws and the unique asymmetries generated by the square-root nonlinearity are emphasized by a comparison with Burgers' equation. These results enable the development and verification of numerical approximations and provide insight into the qualitative dynamics of propagating waves