Similarity reductions and wave solutions for the 3D-Kudryashov–Sinelshchikov equation with variable-coefficients in gas bubbles for a liquid

The variable-coefficients 3D-Kudryashov–Sinelshchikov (vcKS) equation is an important mathematical model that describes the pressure of gas bubbles in the liquid, and is considered as an impressive phenomenon with many applications in medicine, ocean dynamics and fluid mechanics. In this paper, the vcKS is investigated via the enlarged direct similarity technique and reduced to a sixth-order nonlinear differential equation. The Jacobi elliptic wave function method is used to construct many novel periodic and solitary wave solutions for the vcKS equation. Additionally, a physical application of bubble-liquid in medicine was given and some new graphical representation for the dynamic behavior of the obtained periodic shock waves and solitonic waves were constructed to show how the different variable functions choices affect on the wave propagation. Moreover, this effect appears like a higher shock wave by increasing independent variables, which is very important in the drug delivery process.