scholarly journals Soliton Molecules and Full Symmetry Groups to the KdV-Sawada-Kotera-Ramani Equation

2021 ◽  
Vol 2021 ◽  
pp. 1-7
Author(s):  
Na Xiong ◽  
Ya-Xuan Yu ◽  
Biao Li
Keyword(s):  
Symmetry Group ◽  
Direct Method ◽  
Dimensional Space ◽  
Invariant Solution ◽  
Soliton Solutions ◽  
General Group ◽  
One Dimensional ◽  
Group Invariant ◽  
Group A ◽  
Full Symmetry

By N -soliton solutions and a velocity resonance mechanism, soliton molecules are constructed for the KdV-Sawada-Kotera-Ramani (KSKR) equation, which is used to simulate the resonances of solitons in one-dimensional space. An asymmetric soliton can be formed by adjusting the distance between two solitons of soliton molecule to small enough. The interactions among multiple soliton molecules for the equation are elastic. Then, full symmetry group is derived for the KSKR equation by the symmetry group direct method. From the full symmetry group, a general group invariant solution can be obtained from a known solution.

Group Invariant Solutions of the Full Plastic Torsion of Rod with Arbitrary Shaped Cross Sections

2012 ◽  
Vol 4 (03) ◽  
pp. 382-388 ◽  
Author(s):  
Kefu Huang ◽  
Houguo Li
Keyword(s):  
Symmetry Group ◽  
Cross Sections ◽  
Lie Group ◽  
Equilibrium Equation ◽  
Yield Criterion ◽  
Invariant Solution ◽  
Invariant Solutions ◽  
Group Invariant ◽  
Group Invariant Solutions ◽  
Full Symmetry

AbstractBased on the theory of Lie group analysis, the full plastic torsion of rod with arbitrary shaped cross sections that consists in the equilibrium equation and the non-linear Saint Venant-Mises yield criterion is studied. Full symmetry group admitted by the equilibrium equation and the yield criterion is a finitely generated Lie group with ten parameters. Several subgroups of the full symmetry group are used to generate invariants and group invariant solutions. Moreover, physical explanations of each group invariant solution are discussed by all appropriate transformations. The methodology and solution techniques used belong to the analytical realm.

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.

Lie subgroups of the symmetry group of the equations describing a nonstationary and isentropic flow: Invariant and partially invariant solutions

1994 ◽  
Vol 72 (7-8) ◽  
pp. 362-374 ◽  
Author(s):  
A. M. Grundland ◽  
L. Lalague
Keyword(s):  
Symmetry Group ◽  
Dimensional Space ◽  
Projective Group ◽  
Invariant Solutions ◽  
Partially Invariant Solutions ◽  
Isentropic Flow ◽  
Dimensional Space Time ◽  
Group A ◽  
Special Case

We study the symmetries of the equations describing a nonstationary and isentropic flow for an ideal and compressible fluid in four-dimensional space-time. We prove that this system of equations is invariant under the Galilean-similitude group. In the special case of the adiabatic exponent γ = 5/3, corresponding to a diatomic gas, the symmetry group of this system is larger. It is invariant under the Galilean-projective group. A representatives list of subalgebras of Galilean similitude and Galilean-projective Lie algebras, obtained by the method of classification by conjugacy classes under the action of their respective Lie groups, is presented. The results are given in a normalized list and summarized in tables. Examples of invariant and nonreducible partially invariant solutions, obtained from this classification, is constructed. The final part of this work contains an analysis of this classification in connection with a further classification of the symmetry algebras for the Euler and magnetohydrodynamics equations.

Symmetry Reductions and Group-Invariant Radial Solutions to the n-Dimensional Wave Equation

2018 ◽  
Vol 73 (2) ◽  
pp. 161-170 ◽  
Author(s):  
Wei Feng ◽  
Songlin Zhao
Keyword(s):  
Wave Equation ◽  
Differential Equations ◽  
Symmetry Group ◽  
Group Method ◽  
Radial Solutions ◽  
Radial Wave ◽  
One Dimensional ◽  
Group Invariant ◽  
Dimensional Wave ◽  
Radial Wave Equation

AbstractIn this paper, we derive explicit group-invariant radial solutions to a class of wave equation via symmetry group method. The optimal systems of one-dimensional subalgebras for the corresponding radial wave equation are presented in terms of the known point symmetries. The reductions of the radial wave equation into second-order ordinary differential equations (ODEs) with respect to each symmetry in the optimal systems are shown. Then we solve the corresponding reduced ODEs explicitly in order to write out the group-invariant radial solutions for the wave equation. Finally, several analytical behaviours and smoothness of the resulting solutions are discussed.

scholarly journals Some General New Einstein Walker Manifolds

2013 ◽  
Vol 2013 ◽  
pp. 1-8 ◽  
Author(s):  
Mehdi Nadjafikhah ◽  
Mehdi Jafari
Keyword(s):  
Partial Differential Equations ◽  
Symmetry Group ◽  
Lie Symmetry ◽  
Group Method ◽  
Invariant Solutions ◽  
Point Symmetry Group ◽  
One Dimensional ◽  
Group Invariant ◽  
Group Invariant Solutions ◽  
Walker Manifold

Lie symmetry group method is applied to find the Lie point symmetry group of a system of partial differential equations that determines general form of four-dimensional Einstein Walker manifold. Also we will construct the optimal system of one-dimensional Lie subalgebras and investigate some of its group invariant solutions.

Exact solutions of the nonlocal Sawada–Kotera equation in the Alice–Bob system

2020 ◽  
Vol 34 (32) ◽  
pp. 2050315
Author(s):  
Wei-Ping Cao ◽  
Jin-Xi Fei ◽  
Sheng-Wan Fan ◽  
Zheng-Yi Ma ◽  
Hui Xu
Keyword(s):  
Exact Solutions ◽  
Time Reversal ◽  
Dynamic Properties ◽  
Invariant Solution ◽  
Soliton Solutions ◽  
Group Invariant ◽  
New Form ◽  
Delayed Time ◽  
Shifted Parity ◽  
Delayed Time Reversal

To find the group invariant solution, a new shifted parity and delayed time reversal symmetries are used in order to construct Alice–Bob systems. With the help of simple assumptions and of the [Formula: see text] symmetry, the intrinsic two-place model of the Sawada–Kotera (SK) system is elucidated. A new form of the [Formula: see text]-soliton solution for the nonlocal SK equation is obtained, and dynamic properties of the [Formula: see text]-soliton solutions with different values are discussed, respectively. In addition, the breather solution for the AB–SK system is also explicitly identified.

FINITE SYMMETRY GROUP AND COHERENT SOLITON SOLUTIONS FOR THE BROER–KAUP–KUPERSHMIDT SYSTEM

2013 ◽  
Vol 23 (09) ◽  
pp. 1350156 ◽  
Author(s):  
JUN YU ◽  
HANWEI HU
Keyword(s):  
Symmetry Group ◽  
Direct Method ◽  
Soliton Solutions ◽  
Mathematical Physics ◽  
Lie Symmetry ◽  
Point Symmetry ◽  
Symmetry Groups ◽  
Localized Structures ◽  
Lie Point Symmetry

A modified CK direct method is generalized to find finite symmetry groups of nonlinear mathematical physics systems. For the (2 + 1)-dimensional Broer–Kaup–Kupershmidt (BKK) system, both the Lie point symmetry and the non-Lie symmetry groups are obtained by this method. While using the traditional Lie approach, one can only find the Lie symmetry groups. Furthermore, abundant localized structures of the BKK equation are also obtained from the non-Lie symmetry group.

Parametric surfaces

1949 ◽  
Vol 45 (1) ◽  
pp. 14-23 ◽  
Author(s):  
A. S. Besicovitch
Keyword(s):  
Dimensional Space ◽  
Parametric Surfaces ◽  
Rectifiable Curve ◽  
One Dimensional ◽  
Dimensional Measure

In 1914 Carathéodory defined m–dimensional measure in n–dimensional space. He considered one-dimensional measure as a generalization of length and he proved that the length of a rectifiable curve coincides with its one-dimensional measure.

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