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 $(2+1)$-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 $(2+1)$-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.

本文研究了非线性振动和色散波系统的交叉扭结和孤立波解。解决方案包括周期性、交叉扭结和孤立波解决方案。双线性形式是根据广田导数来考虑的。因此，我们利用 Cole-Hopf 算法来获得$(2+1)$维修正色散水波系统。研究了交叉扭结、孤立波解的解析处理，并以二维、密度和三维图的三种形式绘制。将研究非线性振动系统。采用适当的数学假设，考虑到交叉扭结的扭结、周期和孤子的组合以及孤立波的指数函数方面两种扭结的组合，推导和构造了新的交叉扭结和孤立波解。的控制方程。为了实现这一点，说明性示例$(2+1)$-D 改进的分散水波系统提供了展示本研究中使用的程序的可行性和可靠性。行波的轨迹解被明确和图形化地呈现。还考虑了自由参数对两个非线性有理精确情况的少数获得解的设计图形行为的影响。通过将建议的技术与其他现有方案进行比较，结果表明该技术的执行简洁、简单和直接。