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In this work, weakly nonlinear wave equation in mono-dispersed, isothermal, bubbly, slightly compressible viscoelastic liquid is derived, using reduction perturbation method. The constitutive equation is based on the second-grade fluid model. Viscosity, relaxation, retardation and surface tension are considered under isothermal condition, which modifies the Rayleigh–Plesset equation for compressible liquid. A kink traveling wave solution is obtained using the tangent hyperbolic method combined with the Riccati equation. Graphical representation of the solution is given and analyzed with different parameter values.

This study is made to extract the exact solutions of Korteweg–de Vries–Burgers (KdVB) equation and Korteweg–de Vries (KdV) equation. The original idea of this work is to investigate KdV equation and KdVB equation for possible closed form solutions by employing the modified auxiliary equation (MAE) method. Exact traveling wave solutions of the considered equations are retrieved in the form of trigonometric and hyperbolic functions. Kink, periodic and singular wave patterns are obtained from the constructed solutions. The graphical illustration of the wave solutions is presented using 3D-surface plots to acquire the understanding of physical behavior of the obtained results up to possible extent.