Dynamics of free surface shock waves for a saturated superfluid helium (4He\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$^{4}He$$\end{document} ) film on a nanopore substrate

We have explored a saturated (∼10-6\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sim 10^{-6}$$\end{document} cm) superfluid 4He\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$^{4}He$$\end{document} film adsorbed on a nano pore substrate material. We have considered a weak and localized superfluid flow into the porous substrate and investigated its effect on the shock wave dynamics at the surface. The free surface wave dynamics of the superfluid film follows forced Korteweg de Vries (KdV) equation. For a specific choice of nontrivial boundary condition at z=0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$z=0$$\end{document}, the nonlinear evolution equation turns into Korteweg de Vries-Burger equation having a shock wave solution. Such kind of analytical treatment using shallow water superfluid hydrodynamics is new in this field and may generate further research.