The nonlinear vibration and dispersive wave systems with cross-kink and solitary wave solutions

This paper investigates the cross-kink and solitary wave solutions to the nonlinear vibration and dispersive wave systems. The solutions include periodic, cross-kink and solitary wave solutions. The bilinear form is considered in terms of Hirota derivatives. Accordingly, we utilize the Cole–Hopf algorithm to get the exact solutions of the [Formula: see text]-dimensional modified dispersive water-wave system. The analytical treatment of cross-kink, solitary wave solutions is studied and plotted in three forms of two, density and three-D plots. A nonlinear vibration system will be investigated. Employing appropriate mathematical assumptions, the novel kinds of the cross-kink and solitary wave solutions are derived and constructed in view of the combination of kink, periodic and soliton for cross-kink and also a combination of two kinks in terms of exponential functions for solitary of the governing equation. To achieve this, the illustrative example of the [Formula: see text]-D modified dispersive water-wave system is furnished to exhibit the feasibility and reliability of the procedure utilized in this research. The trajectory solutions of the traveling waves are presented explicitly and graphically. The effect of the free parameters on the behavior of designed figures of a few obtained solutions for two nonlinear rational exact cases was also considered. By comparing the suggested technique with the other existing schemes, the results present that the execution of this technique is concise, simple and straightforward.

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