Analytical Solutions and Analyses for the Deflection of Nonlinear Waves on Kirchhoff Plates Underlying a Pasternak-like Nonlinear Elastic Foundation

The present study models the deflection of nonlinear waves over a Kirchhoff plate underlying a Pasternak-like elastic foundation. A promising version of the tanh expansion analytical method has been deployed for the construction of regular exact solutions for the model, including the application of certain ansatz functions for validations and yet construction of more solutions. The resulting frequency equation and the modulation instability spectrum have been obtained for the linearized model, including the expressions for the related phase and group velocities. In addition, the study examines the equilibrium status of the resulting dynamical system with the help of the bifurcation analysis. Numerically, nonlinear deflection and dispersion of waves have been simulated through the acquired expressions and equations. Notably, the study notes that increasing both the Pasternak-like nonlinear parameter η and time variation (for x>0) decreases the nonlinear deflection in the plate, while increasing the stiffness of the Winkler foundation increases deflection in the medium. In addition, the study establishes, concerning the determined frequency equation, that increasing the Winkler foundation stiffness increases the dispersion of nonlinear waves in the medium, while an opposite trend has been noted concerning the imposed Pasternak-like nonlinear foundation. In addition, both phase and group velocities, the gain function for modulation instability, and the resulting dynamical system have been noted to be greatly affected by the variation of the imposed foundational parameters. Lastly, this study has potential applications in various engineering fields while modeling and analysis of mechanical structures supported by additional structures.