Multi-soliton solutions and the Hirota direct technique in the context of the Caudrey-Dodd-Gibbon equation
Keywords:
Soliton solutions, multi-soliton solutions, the Hirota direct method, the Caudrey-Dodd-Gibbon equation, D-operatorAbstract
In this study, we investigate the multi-soliton solutions of the Caudrey-Dodd-Gibbon (CDG) equation using the Hirota direct method, which is a powerful tool for the analysis of integrable nonlinear evolution equations. Many studies have been conducted on soliton dynamics in integrable systems; the study of multi-soliton interactions in the CDG equation is still underdeveloped. The Hirota bilinear transformation is used to convert the nonlinear CDG equation into a bilinear form using the Hirota -operator. Through perturbative expansion, explicit expressions for one-, two-, and three-soliton solutions are systematically obtained. The analysis is extended to formulate a general -soliton solution in closed form, satisfying the multi-soliton condition. The dynamic properties of the obtained solutions are examined through two- and three-dimensional graphical representations, which exhibit the effects of various key parameters on wave propagation, amplitude, and wavelength. We consider several cases, including the appearance of a single soliton and the change in propagation direction under different parameter values. The results reveal the complex wave interactions and phase shifts that are characteristic of nonlinear wave systems. This study highlights the simplicity and efficiency of the Hirota method in generating multi-soliton solutions for high-order nonlinear equations, which provides valuable insights for applications in fluid dynamics, nonlinear optics, and other areas of mathematical physics.
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