VARIABLE-COEFFICIENT FORCED BURGERS SYSTEM IN NONLINEAR FLUID MECHANICS AND ITS POSSIBLY OBSERVABLE EFFECTS

2003 ◽  
Vol 14 (09) ◽  
pp. 1207-1222 ◽  
Author(s):  
YI-TIAN GAO ◽  
XIAO-GE XU ◽  
BO TIAN
Keyword(s):  
Variable Coefficient ◽  
Burgers Equation ◽  
Analytic Solutions ◽  
Charge Density Waves ◽  
Hyperbolic Function ◽  
Disordered Solids ◽  
Exact Analytic Solutions ◽  
Hyperbolic Function Method ◽  
Wu Method ◽  
Nonlinear Fluid

The forced Burgers equation works as a testing ground for a real turbulence, and as the qualitative model for a wide variety of problems including charge density waves, vortex lines in superconductors, disordered solids and epitaxial growth, etc. Its variable-coefficient generalizations call for better modeling of the physical situations. In this paper, we investigate a variable-coefficient generalization of the forced Burgers equation, and obtain several sets of exact soliton-like and other exact analytic solutions, via the extension of a generalized hyperbolic-function method with computerized symbolic computation. We also discuss the Wu method. We find some possibly observable effects, which might be discovered with the relevant experiments.

ON A VARIABLE-COEFFICIENT MODIFIED KP EQUATION AND A GENERALIZED VARIABLE-COEFFICIENT KP EQUATION WITH COMPUTERIZED SYMBOLIC COMPUTATION

2001 ◽  
Vol 12 (06) ◽  
pp. 819-833 ◽  
Author(s):  
YI-TIAN GAO ◽  
BO TIAN
Keyword(s):  
Symbolic Computation ◽  
Variable Coefficient ◽  
Evolution Equations ◽  
Nonlinear Evolution ◽  
Analytic Solutions ◽  
Nonlinear Evolution Equations ◽  
Hyperbolic Function ◽  
Kp Equation ◽  
Exact Analytic Solutions ◽  
Hyperbolic Function Method

The variable-coefficient nonlinear evolution equations, although realistically modeling various mechanical and physical situations, often cause some well-known powerful methods not to work efficiently. In this paper, we extend the power of the generalized hyperbolic-function method, which is based on the computerized symbolic computation, to a variable-coefficient modified Kadomtsev–Petviashvili (KP) equation and a generalized variable-coefficient KP equation. New exact analytic solutions thus come out.

SYMBOLIC COMPUTATION AND POSSIBLY OBSERVABLE EFFECTS FOR THE SOLITON-LIKE LIQUID WAVE PROPAGATION IN THE PRESENCE OF SURFACE TENSION

2004 ◽  
Vol 15 (04) ◽  
pp. 545-551 ◽  
Author(s):  
BO TIAN ◽  
YI-TIAN GAO
Keyword(s):  
Surface Tension ◽  
Wave Propagation ◽  
Symbolic Computation ◽  
Analytic Solutions ◽  
Hyperbolic Function ◽  
Liquid Propellant ◽  
Order Model ◽  
Exact Analytic Solutions ◽  
Hyperbolic Function Method ◽  
Wu Method

Phenomena on the liquid surfaces are of current importance, such as those for rivers, oceans, liquid propellant for rockets and aviation kerosene. Wave propagation on a liquid surface in the presence of surface tension is hereby investigated. Computerized symbolic computation, generalized hyperbolic-function method and Wu method are combined. Some soliton-like, exact analytic solutions are obtained for a recently-proposed (2+1)-dimensional 6th-order model. Possibly observable effects are also proposed, which certain future experiments may hopefully discover.

scholarly journals Improved hyperbolic function method and exact solutions for variable coefficient Benjamin-Bona-Mahony-Burgers equation

2015 ◽  
Vol 19 (4) ◽  
pp. 1183-1187
Author(s):  
Hong-Cai Ma ◽  
Xiao-Fang Peng ◽  
Dan-Dan Yao
Keyword(s):  
Fluid Mechanics ◽  
Exact Solutions ◽  
Variable Coefficient ◽  
Function Method ◽  
Burgers Equation ◽  
Periodic Wave ◽  
Hyperbolic Function ◽  
Hyperbolic Function Method

By using the improved hyperbolic function method, we investigate the variable coefficient Benjamin-Bona-Mahony-Burgers equation which is very important in fluid mechanics. Some exact solutions are obtained. Under some conditions, the periodic wave leads to the kink-like wave.

scholarly journals Application of the Homotopy Analysis Method for Solving the Variable Coefficient KdV-Burgers Equation

2014 ◽  
Vol 2014 ◽  
pp. 1-4 ◽  
Author(s):  
Dianchen Lu ◽  
Jie Liu
Keyword(s):  
Homotopy Analysis Method ◽  
Variable Coefficient ◽  
Burgers Equation ◽  
Approximate Solutions ◽  
Variable Coefficients ◽  
Hyperbolic Function ◽  
Analysis Method ◽  
Function Form ◽  
Transform Method ◽  
Homotopy Analysis

The homotopy analysis method is applied to solve the variable coefficient KdV-Burgers equation. With the aid of generalized elliptic method and Fourier’s transform method, the approximate solutions of double periodic form are obtained. These solutions may be degenerated into the approximate solutions of hyperbolic function form and the approximate solutions of trigonometric function form in the limit cases. The results indicate that this method is efficient for the nonlinear models with the dissipative terms and variable coefficients.

VARIABLE-COEFFICIENT BALANCING-ACT ALGORITHM EXTENDED TO A VARIABLE-COEFFICIENT MKP MODEL FOR THE ROTATING FLUIDS

2001 ◽  
Vol 12 (09) ◽  
pp. 1383-1389 ◽  
Author(s):  
YI-TIAN GAO ◽  
BO TIAN
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Traveling Waves ◽  
Variable Coefficient ◽  
Large Scale ◽  
Analytic Solutions ◽  
Rotating Fluids ◽  
New Family ◽  
Exact Analytic Solutions ◽  
Scale Motion

The modified Kadomtsev–Petviashvili (MKP) models describe the large-scale motion of such rotating fluids as the atmosphere and oceans. Based on the computerized symbolic computation, we in this paper extend the power of the variable-coefficient balancing-act method, which is recently proposed, to a variable-coefficient MKP model. The model is re-written as a coupled set of partial differential equations, and the algorithm is re-written correspondingly. We obtain a new family of the soliton-like, exact analytic solutions, beyond the traveling waves.

ON THE NONINTEGRABLE HIROTA–SATSUMA SYSTEM OF COUPLED KdV EQUATIONS IN FLUID DYNAMICS

2001 ◽  
Vol 12 (10) ◽  
pp. 1425-1430 ◽  
Author(s):  
YI-TIAN GAO ◽  
BO TIAN ◽  
GUANG-MEI WEI
Keyword(s):  
Fluid Dynamics ◽  
Nonlinear Optics ◽  
Variable Coefficient ◽  
Analytic Solutions ◽  
Dispersion Relations ◽  
Long Waves ◽  
Kdv Equations ◽  
Exact Analytic Solutions ◽  
De Vries ◽  
Coupled Kdv Equations

The coupled Korteweg-de Vries (KdV) equations are important in many fields, such as nonlinear optics, fluid dynamics, supersymmetry and so on. In this paper, we modify the recently proposed variable-coefficient balancing-act algorithm, and thus obtain some exact analytic solutions for the nonintegrable Hirota–Satsuma system of coupled KdV equations, which in fluid dynamics describe interactions of two long waves with different dispersion relations.

POSITONIC SOLUTIONS OF A VARIABLE-COEFFICIENT KdV EQUATION IN FLUID DYNAMICS AND PLASMA PHYSICS

2001 ◽  
Vol 12 (10) ◽  
pp. 1431-1437 ◽  
Author(s):  
YI-TIAN GAO ◽  
BO TIAN
Keyword(s):  
Fluid Dynamics ◽  
Plasma Physics ◽  
Variable Coefficient ◽  
Kdv Equation ◽  
Analytic Solutions ◽  
Exact Analytic Solutions ◽  
De Vries

Positons are a new concept in nonlinear sciences. In this paper, we report some positonic, exact analytic solutions for a variable-coefficient Korteweg-de Vries (KdV) equation arising from fluid dynamics and plasma physics.

CRYSTALLINE GRAVITY

2009 ◽  
Vol 24 (18n19) ◽  
pp. 3243-3255 ◽  
Author(s):  
GERARD 't HOOFT
Keyword(s):  
Finite Number ◽  
Lorentz Group ◽  
Degrees Of Freedom ◽  
Holonomy Group ◽  
Discrete Subgroup ◽  
Analytic Solutions ◽  
Space Time ◽  
Exact Analytic Solutions ◽  
Finite Domains ◽  
Dimensional Crystal

Matter interacting classically with gravity in 3+1 dimensions usually gives rise to a continuum of degrees of freedom, so that, in any attempt to quantize the theory, ultraviolet divergences are nearly inevitable. Here, we investigate a theory that only displays a finite number of degrees of freedom in compact sections of space-time. In finite domains, one has only exact, analytic solutions. This is achieved by limiting ourselves to straight pieces of string, surrounded by locally flat sections of space-time. Next, we suggest replacing in the string holonomy group, the Lorentz group by a discrete subgroup, which turns space-time into a 4-dimensional crystal with defects.

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