Asymptotic behavior of traveling wave solutions of the equations for the flow of a fluid with small viscosity and capillarity

We study the oscillations of the traveling wave solutions of \[ \left \{ {_{{v_t} = - p{{\left ( u \right )}_x} + \epsilon {v_{xx}} - \delta {u_{xxx}}}^{{u_t} = {v_x},}} \right .\] for small $\epsilon$ and $\delta$. These solutions give information about the structure of the shock layers in fluids with small viscosity and capillarity. We conclude that the traveling wave has oscillations with increasing amplitude when $\epsilon$ and $\delta$ approach zero such that $\delta \ne O\left ( {{\epsilon ^2}} \right )$. When $\delta = o\left ( {{\epsilon ^2}} \right )$, if there are oscillations, their amplitude decreases to zero as $\epsilon$ and $\delta$ approach zero. When $\delta = {\epsilon ^2}$ the shape of the traveling wave is independent of the magnitude of $\epsilon$ and $\delta$.