Applications of OHAM and MOHAM for Fractional Seventh-Order SKI Equations

In this article, a comparative study between optimal homotopy asymptotic method and multistage optimal homotopy asymptotic method is presented. These methods will be applied to obtain an approximate solution to the seventh-order Sawada-Kotera Ito equation. The results of optimal homotopy asymptotic method are compared with those of multistage optimal homotopy asymptotic method as well as with the exact solutions. The multistage optimal homotopy asymptotic method relies on optimal homotopy asymptotic method to obtain an analytic approximate solution. It actually applies optimal homotopy asymptotic method in each subinterval, and we show that it achieves better results than optimal homotopy asymptotic method over a large interval; this is one of the advantages of this method that can be used for long intervals and leads to more accurate results. As far as the authors are aware that multistage optimal homotopy asymptotic method has not been yet used to solve fractional partial differential equations of high order, we have shown that this method can be used to solve these problems. The convergence of the method is also addressed. The fractional derivatives are described in the Caputo sense.

Lie symmetry analysis, optimal system and conservation law of a generalized (2+1)-dimensional Hirota–Satsuma–Ito equation

In this paper, a generalized (2+1)-dimensional Hirota–Satsuma–Ito (GHSI) equation is investigated using Lie symmetry approach. Infinitesimal generators and symmetry groups of this equation are presented, and the optimal system is given with adjoint representation. Based on the optimal system, some symmetry reductions are performed and some similarity solutions are provided, including soliton solutions and periodic solutions. With Lagrangian, it is shown that the GHSI equation is nonlinearly self-adjoint. By means of the Lie point symmetries and nonlinear self-adjointness, the conservation laws are constructed. Furthermore, some physically meaningful solutions are illustrated graphically with suitable choices of parameters.

Analytical Treatment of the Generalized Hirota-Satsuma-Ito Equation Arising in Shallow Water Wave

In the current study, an analytical treatment is studied starting from the 2 + 1 -dimensional generalized Hirota-Satsuma-Ito (HSI) equation. Based on the equation, we first establish the evolution equation and obtain rational function solutions by means of the bilinear form with the help of the Hirota bilinear operator. Then, by the suggested method, the periodic, cross-kink wave solutions are also obtained. Also, the semi-inverse variational principle (SIVP) will be utilized for the generalized HSI equation. Two major cases were investigated from two different techniques. Moreover, the improved tan χ ξ method on the generalized Hirota-Satsuma-Ito equation is probed. The 3D, density, and contour graphs illustrating some instances of got solutions have been demonstrated from a selection of some suitable parameters. The existing conditions are handled to discuss the available got solutions. The current work is extensively utilized to report plenty of attractive physical phenomena in the areas of shallow water waves and so on.

Cross-kink wave, solitary, dark, and periodic wave solutions by bilinear and He’s variational direct methods for the KP–BBM equation

This paper deals with cross-kink waves in the (2+1)-dimensional KP–BBM equation in the incompressible fluid. Based on Hirota’s bilinear technique, cross-kink solutions related to KP–BBM equation are constructed. Taking the special reduction, the exact expression of different types of solutions comprising exponential, trigonometric and hyperbolic functions is obtained. Moreover, He’s variational direct method (HVDM) based on the variational theory and Ritz-like method is employed to construct the abundant traveling wave solutions of the (2+1)-dimensional generalized Hirota–Satsuma–Ito equation. These traveling wave solutions include kinky dark solitary wave solution, dark solitary wave solution, bright solitary wave solution, periodic wave solution and so on, which are all depending on the initial hypothesis for the Ritz-like method. In continuation, the modulation instability is engaged to discuss the stability of the obtained solutions. Moreover, the rational [Formula: see text] method on the generalized Hirota–Satsuma–Ito equation is investigated. The applicability and effectiveness of the acquired solutions are presented through the numerical results in the form of 3D and 2D graphs. A variety of interactions are illustrated analytically and graphically. The influence of parameters on propagation is analyzed and summarized. The results and phenomena obtained in this paper enrich the dynamic behavior of the evolution of nonlinear waves.

Lump solutions to a generalized Kadomtsev–Petviashvili–Ito equation

A (2+1)-dimensional generalized Kadomtsev–Petviashvili–Ito equation is introduced. Upon adding some second-order derivative terms, its various lump solutions are explicitly constructed by utilizing the Hirota bilinear method and calculated through the symbolic computation system Maple. Furthermore, two specific lump solutions are obtained with particular choices of the parameters and their dynamical behaviors are analyzed through three-dimensional plots and contour plots.

Computational solutions of the generalized Ito equation in nonlinear dispersive systems

This papers presents new exact analytical solutions of a generalized Ito equation having three nonlinear terms, third- and fifth-order derivative forms that model the dynamics of traveling waves in nonlinear dispersive systems. With the help of Riccati equation method, we obtain different kinds of exact traveling wave solutions containing dark, singular, trigonometric, rational and other form of waves solutions that are more general than classical ones existing in the literature. Despite the originality of the new results obtained, the method used here is very efficient, powerful and can be extended to other types of nonlinear equations and more. Moreover, the behaviors of traveling waves solutions are portrayed graphically by selecting suitable values for the physical parameters.