Solitary waves in magnetostrictive circular rods with temperature effect

2021 ◽  
Author(s):  
H. T. Jia ◽  
Chun-Xia Xue ◽  
Q. Chen
Keyword(s):  
Solitary Wave ◽  
Wave Velocity ◽  
Solitary Wave Solution ◽  
Kdv Equation ◽  
Constitutive Relations ◽  
Velocity Ratio ◽  
Expansion Method ◽  
Reductive Perturbation Method ◽  
Normal Strain ◽  
Nonlinear Superposition

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.

Solitary and freak waves in superthermal plasma with ion jet

2012 ◽  
Vol 79 (3) ◽  
pp. 287-294 ◽  
Author(s):  
U. M. ABDELSALAM
Keyword(s):  
Solitary Wave Solution ◽  
Kdv Equation ◽  
Ion Beam ◽  
Negative Ions ◽  
Reductive Perturbation Method ◽  
Dust Particles ◽  
Superthermal Electrons ◽  
Plasma Parameters ◽  
Freak Waves ◽  
Superthermal Plasma

AbstractThe nonlinear solitary and freak waves in a plasma composed of positive and negative ions, superthermal electrons, ion beam, and stationary dust particles have been investigated. The reductive perturbation method is used to obtain the Korteweg-de Vries (KdV) equation describing the system. The latter admits solitary wave solution, while the dynamics of the modulationally unstable wavepackets described by the KdV equation gives rise to the formation of freak/rogue excitation described by the nonlinear Schrödinger equation. In order to show that the characteristics of solitary and freak waves are influenced by plasma parameters, relevant numerical analysis of appropriate nonlinear solutions are presented. The results from this work predict nonlinear excitations that may associate with ion jet and superthermal electrons in Herbig–Haro objects.

scholarly journals A solitary-wave solution to a perturbed KdV equation

2000 ◽  
Vol 64 (4) ◽  
pp. 475-480 ◽  
Author(s):  
M. A. ALLEN ◽  
G. ROWLANDS
Keyword(s):  
Solitary Wave ◽  
Wave Solution ◽  
Solitary Wave Solution ◽  
Kdv Equation ◽  
Perturbation Expansion ◽  
Approximate Form ◽  
First Order Correction ◽  
Higher Order Corrections ◽  
Perturbed Kdv Equation ◽  
Secular Terms

We derive the approximate form and speed of a solitary-wave solution to a perturbed KdV equation. Using a conventional perturbation expansion, one can derive a first-order correction to the solitary-wave speed, but at the next order, algebraically secular terms appear, which produce divergences that render the solution unphysical. These terms must be treated by a regrouping procedure developed by us previously. In this way, higher-order corrections to the speed are obtained, along with a form of solution that is bounded in space. For this particular perturbed KdV equation, it is found that there is only one possible solitary wave that has a form similar to the unperturbed soliton solution.

Study of the Solitoff Solution and Fractal Solution for the (2+1)-Dimensional Broek-Kaup System with Variable Coefficients

2014 ◽  
Vol 532 ◽  
pp. 356-361
Author(s):  
Wei Ting Zhu
Keyword(s):  
Solitary Wave ◽  
Exact Solutions ◽  
Wave Solution ◽  
Solitary Wave Solution ◽  
Expansion Method ◽  
Variable Coefficients ◽  
Variable Separation ◽  
New Family ◽  
Localized Excitations ◽  
Variable Separation Method

Starting from a (G'/G)-expansion method and a variable separation method, a new family of exact solutions of the (2+1)-dimensional Broek-Kaup system with variable coefficients(VCBK) is obtained. Based on the derived solitary wave solution, we obtain some special localized excitations such as solitoff solutions and fractal solutions.

THE STABILITY OF THE SOLITARY WAVE SOLUTIONS TO THE GENERALIZED COMPOUND KDV EQUATION

2011 ◽  
Vol 04 (03) ◽  
pp. 475-480
Author(s):  
Xiaohua Liu ◽  
Weiguo Zhang
Keyword(s):  
Solitary Wave ◽  
Variational Method ◽  
Wave Solution ◽  
Lyapunov Functional ◽  
Solitary Wave Solution ◽  
Kdv Equation ◽  
Solitary Wave Solutions ◽  
Wave Solutions ◽  
The Stability ◽  
Lyapunov Sense

Using variational method, we investigate that the solitary wave solution u(x - ct) to the Generalized Compound Kdv Equation with two nonlinear terms is stable in the Lyapunov sense when 0 < p < 2 holds. The result is new. There shows a new method to consider the extremum of Lyapunov functional.

Exact Periodic Solitary-Wave Solution for KdV Equation

2008 ◽  
Vol 25 (5) ◽  
pp. 1531-1533 ◽  
Author(s):  
Dai Zheng-De ◽  
Liu Zhen-Jiang ◽  
Li Dong-Long
Keyword(s):  
Solitary Wave ◽  
Wave Solution ◽  
Solitary Wave Solution ◽  
Kdv Equation

On asymmetric gravity–capillary solitary waves

1997 ◽  
Vol 330 ◽  
pp. 215-232 ◽  
Author(s):  
T.-S. YANG ◽  
T. R. AKYLAS
Keyword(s):  
Solitary Wave ◽  
Solitary Waves ◽  
Numerical Solutions ◽  
Solitary Wave Solution ◽  
Kdv Equation ◽  
Phase Speed ◽  
Capillary Waves ◽  
Asymptotic Results ◽  
Water Wave Problem ◽  
Fifth Order

Symme tric gravity–capillary solitary waves with decaying oscillatory tails are known to bifurcate from infinitesimal periodic waves at the minimum value of the phase speed where the group velocity is equal to the phase speed. In the small-amplitude limit, these solitary waves may be interpreted as envelope solitons with stationary crests and are described by the nonlinear Schrödinger (NLS) equation to leading order. In line with this interpretation, it would appear that one may also co nstruct asymmetric solitary waves by shifting the carrier oscillations relative to the envelope of a symmetric solitary wave. This possibility is examined here on the basis of the fifth-order Korteweg–de Vries (KdV) equation, a model for g ravity–capillary waves on water of finite depth when the Bond number is close to 1/3. Using techniques of exponential asymptotics beyond all orders of the NLS theory, it is shown that asymmetric solitary waves of the form suggested by the NLS theory in fact are not possible. On the other hand, an infinity of symmetric and asymmetric solitary-wave solution families comprising two or more NLS solitary wavepackets bifurcate at finite values of the amplitude parameter. The asymptotic results are consistent with numerical solutions of the fifth-order KdV equation. Moreover, the asymptotic theory suggests that such multi-packet gravity–capillary solitary waves also exist in the full water-wave problem near the minimum of t he phase speed.

Structure-preserving analysis on Gaussian solitary wave solution of logarithmic-KdV equation

2021 ◽  
Author(s):  
Jingjing Hu ◽  
Weipeng Hu ◽  
Fan Zhang ◽  
Han Zhang ◽  
Zichen Deng
Keyword(s):  
Solitary Wave ◽  
Symmetry Breaking ◽  
Conservation Laws ◽  
Wave Solution ◽  
Solitary Wave Solution ◽  
Kdv Equation ◽  
Dynamic Symmetry ◽  
Structure Preserving ◽  
Numerical Errors ◽  
Simulation Process

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.

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