Higher Order Solution of Nonlinear Waves. II. Shock Wave Described by Burgers Equation

The concept of ranked order probability distribution unveils natural probabilistic interpretations for the kink waves (and hence the solitons) solving higher order dispersive Burgers’ type PDEs. Thanks to this underlying structure, it is possible to propose a systematic derivation of exact solutions for PDEs with a quadratic nonlinearity of the Burgers’ type but with arbitrary dispersive orders. As illustrations, we revisit the dissipative Kotrweg de Vries, Kuramoto-Sivashinski, and Kawahara equations (involving third, fourth, and fifth order dispersion dynamics), which in this context appear to be nothing but the simplest special cases of this infinitely rich class of nonlinear evolutions.

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Higher‐order solution of an ion‐acoustic solitary wave in a plasma

Physics of Fluids B Plasma Physics ◽

10.1063/1.860526 ◽

1993 ◽

Vol 5 (2) ◽

pp. 409-414 ◽

Cited By ~ 23

Author(s):

S. Watanabe ◽

B. Jiang

Keyword(s):

Solitary Wave ◽

Higher Order ◽

Order Solution ◽

Ion Acoustic

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Regularization of the Shock Wave Solution to the Riemann Problem for the Relativistic Burgers Equation

Abstract and Applied Analysis ◽

10.1155/2014/178672 ◽

2014 ◽

Vol 2014 ◽

pp. 1-10

Author(s):

Ting Zhang ◽

Chun Shen

Keyword(s):

Shock Wave ◽

Initial Data ◽

Riemann Problem ◽

Wave Solution ◽

Burgers Equation ◽

Explicit Computation ◽

Convective Term ◽

Regularization Technique ◽

Shock Wave Solution

The regularization of the shock wave solution to the Riemann problem for the relativistic Burgers equation is considered. For Riemann initial data consisting of a single decreasing jump, we find that the regularization of nonlinear convective term cannot capture the correct shock wave solution. In order to overcome it, we consider a new regularization technique called the observable divergence method introduced by Mohseni and discover that it can capture the correct shock wave solution. In addition, we take the Helmholtz filter for the fully explicit computation.

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Exact statistical properties of the Burgers equation

Journal of Fluid Mechanics ◽

10.1017/s0022112000001142 ◽

2000 ◽

Vol 417 ◽

pp. 323-349 ◽

Cited By ~ 52

Author(s):

L. FRACHEBOURG ◽

Ph. A. MARTIN

Keyword(s):

White Noise ◽

Large Distance ◽

Burgers Equation ◽

Higher Order ◽

Statistical Properties ◽

Inviscid Limit ◽

Initial Condition ◽

Spatial Correlations ◽

One Dimensional ◽

The One

The one-dimensional Burgers equation in the inviscid limit with white noise initial condition is revisited. The one- and two-point distributions of the Burgers field as well as the related distributions of shocks are obtained in closed analytical forms. In particular, the large distance behaviour of spatial correlations of the field is determined. Since higher-order distributions factorize in terms of the one- and two- point functions, our analysis provides an explicit and complete statistical description of this problem.

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Classification and Recursion Operators of Dark Burgers’ Equation

Zeitschrift für Naturforschung A ◽

10.1515/zna-2017-0324 ◽

2018 ◽

Vol 73 (2) ◽

pp. 175-180 ◽

Cited By ~ 5

Author(s):

Mei-Dan Chen ◽

Biao Li

Keyword(s):

Symbolic Computation ◽

Free Parameter ◽

Burgers Equation ◽

Differential Polynomial ◽

Higher Order ◽

Recursion Operators ◽

Burgers Equations ◽

Dual Symmetry ◽

Free Parameters

AbstractWith the help of symbolic computation, two types of complete scalar classification for dark Burgers’ equations are derived by requiring the existence of higher order differential polynomial symmetries. There are some free parameters for every class of dark Burgers’ systems; so some special equations including symmetry equation and dual symmetry equation are obtained by selecting the free parameter. Furthermore, two kinds of recursion operators for these dark Burgers’ equations are constructed by two direct assumption methods.

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