On the propagation of nonlinear SH waves in a two-layered compressible elastic medium

We examine nonlinear shear horizontal (SH) waves in a two-layered medium of uniform thickness. Materials in both layers having different material properties are assumed to be nonlinear elastic, homogeneous, compressible and isotropic. Furthermore, c1<c2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$c_1<c_2$$\end{document} is chosen where c1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$c_1$$\end{document} and c2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$c_2$$\end{document} are the linear shear wave velocities of the top and bottom layers, respectively. The propagation of SH waves in a two-layered medium exists only if either of the inequalities c1<c<c2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$c_1<c< c_2$$\end{document} and c1<c2<c\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$c_1<c_2< c$$\end{document} is satisfied where c refers to the phase velocity of waves. The dispersion relations of linear SH waves corresponding to these phase inequalities are obtained. Then, the self-modulation of nonlinear SH waves is investigated by using an asymptotic perturbation method for each case. By balancing dispersion and weak nonlinearity, it is shown that the self-modulation of the first-order slowly varying amplitude of nonlinear SH waves is characterized asymptotically by nonlinear Shrödinger (NLS) equation. The form of NLS equations corresponding to each phase inequality is identical except their coefficients. Then, the effects of nonlinearity of the materials in layers on the linear stability of the solution of NLS equation and on the existence of solitary wave solutions are studied by employing several fictive material parameter models.