scholarly journals Trivial conservation laws and solitary wave solution of the fifth order Lax equation

2021 ◽  
pp. 100101
Author(s):  
Arzu Akbulut ◽  
Filiz Taşcan ◽  
Elif Özel
Keyword(s):  
Solitary Wave ◽  
Conservation Laws ◽  
Wave Solution ◽  
Solitary Wave Solution ◽  
Lax Equation ◽  
Fifth Order

Solitary wave solution to a class of higher-order viscous traffic flow model

2017 ◽  
Vol 31 (13) ◽  
pp. 1750099 ◽  
Author(s):  
Chun-Xiu Wu ◽  
Peng Zhang
Keyword(s):  
Solitary Wave ◽  
Traffic Flow ◽  
Wave Solution ◽  
Solitary Wave Solution ◽  
Higher Order ◽  
Order Accuracy ◽  
Flow Models ◽  
Weighted Essentially Nonoscillatory ◽  
Traffic Flow Models ◽  
Fifth Order

Traveling waves of a class of higher-order traffic flow models with viscosity are studied with the reduction perturbation method, which leads to the well-known Kortweg–de Vries equation and the approximate solitary wave solution to the model. The fifth-order accuracy weighted essentially nonoscillatory scheme is adopted for comparison between the analytical and numerical results. The numerical tests show that the solitary wave evolves with little deformation of its profile and that a globally perturbed equilibrium traffic state is able to evolve into a profile similar to that of a solitary wave, which is identified by the same total number of vehicles on the ring road. These results are compared with those in the literature and demonstrate that the approximation to the model is more accurate.

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.

scholarly journals Abundant Explicit and Exact Solutions for the Variable Coefficient mKdV Equations

2013 ◽  
Vol 2013 ◽  
pp. 1-7 ◽  
Author(s):  
Xiaoxiao Zheng ◽  
Yadong Shang ◽  
Yong Huang
Keyword(s):  
Solitary Wave ◽  
Elliptic Function ◽  
Constant Coefficient ◽  
Wave Solution ◽  
Integral Method ◽  
Solitary Wave Solution ◽  
First Integral ◽  
Mkdv Equation ◽  
Variable Coefficients ◽  
First Integral Method

This paper is concerned with the variable coefficients mKdV (VC-mKdV) equation. First, through some transformation we convert VC-mKdV equation into the constant coefficient mKdV equation. Then, using the first integral method we obtain the exact solutions of VC-mKdV equation, such as rational function solutions, periodic wave solutions of triangle function, bell-shape solitary wave solution, kink-shape solitary wave solution, Jacobi elliptic function solutions, and Weierstrass elliptic function solution. Furthermore, with the aid of Mathematica, the extended hyperbolic functions method is used to establish abundant exact explicit solution of VC-mKdV equation. By the results of the equation, the first integral method and the extended hyperbolic function method are extended from the constant coefficient nonlinear evolution equations to the variable coefficients nonlinear partial differential equation.

A regularized model for strongly nonlinear internal solitary waves

2009 ◽  
Vol 629 ◽  
pp. 73-85 ◽  
Author(s):  
WOOYOUNG CHOI ◽  
RICARDO BARROS ◽  
TAE-CHANG JO
Keyword(s):  
Stability Analysis ◽  
Solitary Wave ◽  
Solitary Waves ◽  
Wave Solution ◽  
Wave Amplitude ◽  
Solitary Wave Solution ◽  
Shear Instability ◽  
Internal Solitary Waves ◽  
Strongly Nonlinear ◽  
Regularized Model

The strongly nonlinear long-wave model for large amplitude internal waves in a two-layer system is regularized to eliminate shear instability due to the wave-induced velocity jump across the interface. The model is written in terms of the horizontal velocities evaluated at the top and bottom boundaries instead of the depth-averaged velocities, and it is shown through local stability analysis that internal solitary waves are locally stable to perturbations of arbitrary wavelengths if the wave amplitudes are smaller than a critical value. For a wide range of depth and density ratios pertinent to oceanic conditions, the critical wave amplitude is close to the maximum wave amplitude and the regularized model is therefore expected to be applicable to the strongly nonlinear regime. The regularized model is solved numerically using a finite-difference method and its numerical solutions support the results of our linear stability analysis. It is also shown that the solitary wave solution of the regularized model, found numerically using a time-dependent numerical model, is close to the solitary wave solution of the original model, confirming that the two models are asymptotically equivalent.

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