scholarly journals Variational theory for (2+1)-dimensional fractional dispersive long wave equations

2021 ◽  
pp. 23-23
Author(s):  
Xiao-Qun Cao ◽  
Cheng-Zhuo Zhang ◽  
Shi-Cheng Hou ◽  
Ya-Nan Guo ◽  
Ke-Cheng Peng
Keyword(s):  
Porous Media ◽  
Nonlinear Waves ◽  
Inverse Method ◽  
Wave Equations ◽  
Continuous Medium ◽  
Variational Principles ◽  
Variational Theory ◽  
Long Wave ◽  
Potential Applications ◽  
Fractional System

This paper extends the (2+1)-dimensional Eckhaus-type dispersive long wave equations in continuous medium to their fractional partner, which is a model of nonlinear waves in fractal porous media. The derivation is shown briefly using He?s fractional derivative. Using the semi-inverse method, the variational principles are established for the fractional system, which up to now are not discovered. The obtained fractal variational principles are proved correct by minimizing the functionals with the calculus of variations, and might find potential applications in numerical modelling.

Variational Principles for Some Nonlinear Wave Equations

2008 ◽  
Vol 63 (5-6) ◽  
pp. 237-240 ◽  
Author(s):  
Zhao-Ling Tao
Keyword(s):  
Nonlinear Wave ◽  
Inverse Method ◽  
Wave Equations ◽  
Boussinesq Equation ◽  
Variational Principles ◽  
Vries Equation ◽  
Nonlinear Wave Equations ◽  
Kawahara Equation ◽  
De Vries

Using the semi-inverse method proposed by Ji-Huan He, variational principles are established for some nonlinear wave equations arising in physics, including the Pochhammer-Chree equation, Zakharov-Kuznetsov equation, Korteweg-de Vries equation, Zhiber-Shabat equation, Kawahara equation, and Boussinesq equation.

A FRACTAL VARIATIONAL THEORY FOR ONE-DIMENSIONAL COMPRESSIBLE FLOW IN A MICROGRAVITY SPACE

Fractals ◽  
2020 ◽  
Vol 28 (02) ◽  
pp. 2050024 ◽  
Author(s):  
JI-HUAN HE
Keyword(s):  
Compressible Flow ◽  
Lagrange Multiplier ◽  
Variational Formulation ◽  
Inverse Method ◽  
Variational Principles ◽  
Microgravity Condition ◽  
Multiplier Method ◽  
Variational Theory ◽  
One Dimensional ◽  
The One

The semi-inverse method is adopted to establish a family of fractal variational principles of the one-dimensional compressible flow under the microgravity condition, and Cauchy–Lagrange integral is successfully derived from the obtained variational formulation. A suitable application of the Lagrange multiplier method is also elucidated.

VARIATIONAL PRINCIPLE FOR (2 + 1)-DIMENSIONAL BROER–KAUP EQUATIONS WITH FRACTAL DERIVATIVES

Fractals ◽  
2020 ◽  
Vol 28 (07) ◽  
pp. 2050107 ◽  
Author(s):  
XIAO-QUN CAO ◽  
SHI-CHENG HOU ◽  
YA-NAN GUO ◽  
CHENG-ZHUO ZHANG ◽  
KE-CHENG PENG
Keyword(s):  
Inverse Method ◽  
Evolution Equations ◽  
Variational Principles ◽  
Nonlinear Evolution ◽  
Nonlinear Evolution Equations ◽  
Conserved Quantities ◽  
Coupled Equations ◽  
Fractal Space ◽  
Potential Applications ◽  
Variational Formulations

This paper extends the [Formula: see text]-dimensional Broer–Kaup equations in continuum mechanics to its fractional partner, which can model a lot of nonlinear waves in fractal porous media. Its derivation is demonstrated in detail by applying He’s fractional derivative. Using the semi-inverse method, two variational principles are established for the nonlinear coupled equations, which up to now are not discovered. The variational formulations can help to study the symmetries and find conserved quantities in the fractal space. The obtained variational principles are proved correct by minimizing the functionals with the calculus of variations, and might find potential applications in numerical simulation. The procedure reveals that the semi-inverse method is highly efficient and powerful, and can be generalized to other nonlinear evolution equations with fractal derivatives.

scholarly journals Variational Principles for Two Kinds of Coupled Nonlinear Equations in Shallow Water

Symmetry ◽  
2020 ◽  
Vol 12 (5) ◽  
pp. 850
Author(s):  
Xiao-Qun Cao ◽  
Ya-Nan Guo ◽  
Shi-Cheng Hou ◽  
Cheng-Zhuo Zhang ◽  
Ke-Cheng Peng
Keyword(s):  
Shallow Water ◽  
Nonlinear Equations ◽  
Wave Equations ◽  
Variational Principles ◽  
Soliton Solutions ◽  
Conserved Quantities ◽  
Long Wave ◽  
Complex Models ◽  
Lagrange Functional ◽  
Variational Formulations

It is a very important but difficult task to seek explicit variational formulations for nonlinear and complex models because variational principles are theoretical bases for many methods to solve or analyze the nonlinear problem. By designing skillfully the trial-Lagrange functional, different groups of variational principles are successfully constructed for two kinds of coupled nonlinear equations in shallow water, i.e., the Broer-Kaup equations and the (2+1)-dimensional dispersive long-wave equations, respectively. Both of them contain many kinds of soliton solutions, which are always symmetric or anti-symmetric in space. Subsequently, the obtained variational principles are proved to be correct by minimizing the functionals with the calculus of variations. The established variational principles are firstly discovered, which can help to study the symmetries and find conserved quantities for the equations considered, and might find lots of applications in numerical simulation.

The Generalisation of Integrable (2+1)-Dimensional Dispersive Long-Wave Equations

1997 ◽  
Vol 66 (5) ◽  
pp. 1288-1290 ◽  
Author(s):  
Thangavel Alagesan ◽  
Ambigapathy Uthayakumar ◽  
Kuppusamy Porsezian
Keyword(s):  
Wave Equations ◽  
Long Wave

scholarly journals Generalized Variational Principle for Long Water-Wave Equation by He's Semi-Inverse Method

2009 ◽  
Vol 2009 ◽  
pp. 1-5 ◽  
Author(s):  
Weimin Zhang
Keyword(s):  
Water Wave ◽  
Inverse Method ◽  
Variational Principles ◽  
Nonlinear Partial Differential Equations ◽  
Nonlinear Partial Differential Equation ◽  
Variational Formula ◽  
Generalized Variational Principle ◽  
Partial Differential ◽  
Water Wave Problem ◽  
Mathematics And Physics

Variational principles for nonlinear partial differential equations have come to play an important role in mathematics and physics. However, it is well known that not every nonlinear partial differential equation admits a variational formula. In this paper, He's semi-inverse method is used to construct a family of variational principles for the long water-wave problem.

Symmetry Reduction, Exact Solutions, and Conservation Laws of the (2+1)-Dimensional Dispersive Long Wave Equations

2009 ◽  
Vol 64 (9-10) ◽  
pp. 597-603 ◽  
Author(s):  
Zhong Zhou Dong ◽  
Yong Chen
Keyword(s):  
Conservation Laws ◽  
Exact Solutions ◽  
Symmetry Group ◽  
Infinite Number ◽  
Wave Equations ◽  
Direct Method ◽  
Discrete Groups ◽  
Point Symmetry Group ◽  
Long Wave ◽  
Full Symmetry

By means of the generalized direct method, we investigate the (2+1)-dimensional dispersive long wave equations. A relationship is constructed between the new solutions and the old ones and we obtain the full symmetry group of the (2+1)-dimensional dispersive long wave equations, which includes the Lie point symmetry group S and the discrete groups D. Some new forms of solutions are obtained by selecting the form of the arbitrary functions, based on their relationship. We also find an infinite number of conservation laws of the (2+1)-dimensional dispersive long wave equations.

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