Construction of traveling wave solutions of the (2 + 1)‐dimensional modified KdV‐KP equation

Abstract

Traveling wave solutions have played a vital role in demonstrating the wave character of nonlinear problems emerging in the field of mathematical sciences and engineering. To depict the nature of propagation of the nonlinear waves in nature, a range of nonlinear evolution equations has been proposed and investigated in the existing literature. In this article, solitary and traveling periodic wave solutions for the (2 + 1)‐dimensional modified KdV‐KP equation are derived by employing an ansatz method, named the enhanced (G′/G)‐expansion method. For this continued equation, abundant solitary wave solutions and nonlinear periodic wave solutions, along with some free parameters, are obtained. We have derived the exact expressions for the solitary waves that arise in the continuum‐modified KdV‐KP model. We study the significance of parameters numerically that arise in the obtained solutions. These parameters play an important role in the physical structure and propagation directions of the wave that characterizes the wave pattern. We discuss the relation between velocity and parameters and illustrate them graphically. Our numerical analysis suggests that the taller solitons are narrower than shorter waves and can travel faster. In addition, graphical representations of some obtained solutions along with their contour plot and wave train profiles are presented. The speed, as well as the profile of these solitary waves, is highly sensitive to the free parameters. Our results establish that the continuum‐modified KdV‐KP system supports solitary waves having different shapes and speeds for different values of the parameters.

Chaotic behavior, bifurcations, sensitivity analysis, and novel optical soliton solutions to the Hamiltonian amplitude equation in optical physics

“…To uncover exact solutions for different nonlinear equations (NLEs) like HAE, researchers employ several methods including the unified method (Fokas andLenells 2012, Abdel-Gawad andOsman 2015), the transformed rational function method (Ma and Lee 2009) the extended direct algebraic method (Gao et al 2020, Shahzad et al 2023, Lie group method (Buckwar andLuchko 1998, Jafari et al 2015), Hirota's bilinear method (Zhou et al 2006, Liu et al 2019, the exp(−j(ξ))-expansion method (Roshid et al 2014, Raza et al 2019, the generalized Kudryshov method (Islam et al 2015, Habib et al 2019, the variational iteration method (Abdel-Gawad and Osman 2013, Seadawy 2015), the Backlund transform method (Yuan et al 2011), the sine-Gordon expansion method (Kumar et al 2017, the Cole-Hopf transformation method (Salas and Gmez S 2010), the auxiliary equation method (Zhang 2013, Islam et al 2023, the homotopy-perturbation method (Cveticanin 2006), the tanh-function method (Parkes andDuffy 1996, Fan 2000), homogeneous balance method (Wang et al 1996, Fan andZhang 1998), Hirota bilinear formulation with N-soliton (Ma 2020 2021), the ( /  G G)-expansion method (Wang et al 2008, Zayed and Gepreel 2009, Khan et al 2023, the fractional approach (Tandel et al 2022), the (G′/G 2 )-expansion method (Islam et al 2022), multiple exp-function method (Ma et al 2010, Ma andZhu 2012), the tanh-coth method Pankaj 2015, Mamun et al 2022) and many others.…”