Stability analysis of linear ion waves leading to nonlinear shock and soliton structures in a dense quantum plasma with strongly coupled ions and degenerate electrons

•The interaction between cold plasma (astrophysical environments) needs to be described. Hence, using MHD mathematical model a theory is presented in this article.•We derived the novel dispersion relation for the dense plasma. We found that the unstable root of the dispersion relation leads to nonlinear wave in dense plasma environment consist of degenerate electron and strongly coupled ions.•We derived relations for the nonlinear shock wave and soliton for given environment. We have explored propagation characteristics of linear and nonlinear ion waves.•Analytical analysis of the expressions for the given environment is presented. The numerical results from expressions have been interpreted in the manuscript.