Approximate Damped Oscillatory Traveling Wave in Viscous Compressible Fluid With Capillary Effect

We convert the viscous compressible fluid equations with capillary effect into the equivalent planar dynamical system and illustrate the global qualitative phase diagrams corresponding to its bounded orbits by the qualitative theory of ordinary differential equations. From the evolutionary relationship of bounded orbits in global qualitative phase diagrams and the correspondence between the bounded orbit and traveling wave, we discover the evolutionary rule of the profile of the traveling wave as the viscous coefficient varying and the forming reason for the damped oscillatory traveling wave. Further, we construct the analytical approximate explicit expression of the damped oscillatory traveling waves in accordance with the forming reason and prove its error decreases in exponential form. Finally, we compare the approximate damped oscillatory traveling wave with the numerical solution. The result shows they fit well.