solitary wave solution
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Author(s):  
Fangcheng Fan

In this paper, we investigate a four-component Toda lattice (TL), which may be used to model the wave propagation in lattices just like the famous TL. By means of the Lax pair and gauge transformation, we construct the [Formula: see text]-fold Darboux transformation (DT), which enables us to obtain multi-soliton or multi-solitary wave solution without complex iterative process. Through the obtained DT, [Formula: see text]-fold explicit exact solutions of the system and their figures with proper parameters are presented from which we find the [Formula: see text]-fold solution shows two-solitary wave structure, the amplitude and shape of the wave change with time. Finally, we derive an infinite number of conservation laws formulaically to illustrate the integrability of the system.


Author(s):  
KangLe Wang

In this work, the Bogoyavlenskii system (BS) and fractal BS are investigated by variational method for the first time. An efficient and simple scheme is proposed to seek their exact solitary wave solutions, which is called variational analysis method. The novel scheme requires only two steps, making it much attractive in practical applications, and a good result is obtained. This paper cleans up the road to the exact solitions, and it sheds a new light on the soliton theory. Finally, the physical properties of solitary wave solutions obtained are analyzed by some simulation figures.


2021 ◽  
Author(s):  
Jingjing Hu ◽  
Weipeng Hu ◽  
Fan Zhang ◽  
Han Zhang ◽  
Zichen Deng

Abstract The existence of the Gaussian solitary wave solution in the logarithmic-KdV equation has aroused considerable interests recently. Focusing on the defects of the reported multi-symplectic analysis on the Gaussian solitary wave solution of the logarithmic-KdV equation and considering the dynamic symmetry breaking of the logarithmic-KdV equation, new approximate multi-symplectic formulations for the logarithmic-KdV equation are deduced and the associated structure-preserving scheme is constructed to simulate the evolution of the Gaussian solitary wave solution. In the structure-preserving simulation process of the Gaussian solitary wave solution, the residuals of three conservation laws are recorded in each step. Comparing with the reported numerical results, it can be found that the Gaussian solitary wave solution can be simulated with tiny numerical errors and three conservation laws are preserved perfectly in the simulation process by the structure-preserving scheme presented in this paper, which implies that the proposed structure-preserving scheme improved the precision as well as the structure-preserving properties of the reported multi-symplectic approach.


2021 ◽  
Vol 153 ◽  
pp. 111495
Author(s):  
Yasir Akbar ◽  
Haleem Afsar ◽  
Fahad S Al-Mubaddel ◽  
Nidal H. Abu-Hamdeh ◽  
Abdullah M. Abusorrah

Author(s):  
H. T. Jia ◽  
Chun-Xia Xue ◽  
Q. Chen

A simple nonlinear model is constructed in this paper to study the solitary wave in an infinite circular magnetostrictive rod. Based on the constitutive relations for transversely isotropic magnetostrictive materials, considering the coupling of multiphysics, combined with Hamilton’s principle and Euler equation, the longitudinal wave equation (LWE) of the infinite circular rod is obtained. The nonlinearity considered is geometrically associated with the nonlinear normal strain in the longitudinal rod direction. The transverse Poisson’s effect is included by introducing the effective Poisson’s ratio. Solitary wave solution, non-topological bell-type soliton and singular periodic solutions of the LWE are obtained by the [Formula: see text]-expansion method. By using the reductive perturbation method, we derive the KdV equation, furthermore, the two-solitary solution is obtained. Numerical analysis results show that the increase of the magnetic field intensity or temperature will reduce the solitary wave’s propagation velocity. As the wave velocity ratio increases, the wave amplitude gradually increases; when the coupled physics parameter and the wave velocity ratio are constant, the increase of the dispersion parameter will make the wavelength longer. The dynamic behavior of the two-soliton solution in the magnetostrictive rod exhibits nonlinear superposition and has elastic collision characteristics.


Author(s):  
Baolin Feng ◽  
Jalil Manafian ◽  
Onur Alp Ilhan ◽  
Amitha Manmohan Rao ◽  
Anand H. Agadi

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.


2021 ◽  
Vol 9 ◽  
Author(s):  
R. Jahangir ◽  
S. Ali

The formation of nonlinear ion-acoustic waves is studied in a degenerate magnetoplasma accounting for quantized and trapped electrons. Relying on the reductive perturbation technique, a three-dimensional Zakharov–Kuznetsov (ZK) equation is derived, admitting a solitary wave solution with modified amplitude and width parameters. The stability of the ZK equation is also discussed using the k-expansion method. Subsequently, numerical analyses are carried out for plasma parameters of a dense stellar system involving white dwarf stars. It has been observed that the quantized magnetic field parameter η and degeneracy of electrons (determined by small temperature values T) affect the amplitude and width of the electric potential. The critical point at which the nature of the solitary structure changes from compressive to rarefaction is evaluated. Importantly, the growth rate of the instability associated with a three-dimensional ZK equation depends on the plasma parameters, and higher values of η and T tend to stabilize the solitons in quantized degenerate plasmas. The results of the present study may hold significance to comprehend the properties of wave propagation and instability growth in stellar and laboratory dense plasmas.


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