Saddle-node bifurcation of viscous profiles

► A saddle-node bifurcation of shock waves in viscous conservation laws is studied. ► The saddle-node bifurcation is described through Melnikov integrals. ► The stability of the bifurcating solutions is studied by the Evans function method. ► The Evans function is analyzed by using geometric singular perturbation theory. ► New connections between derivatives of Evans and Melnikov functions are established.