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.

scholarly journals Peakon Solutions of Alice-Bob b-Family Equation and Novikov Equation

2019 ◽  
Vol 2019 ◽  
pp. 1-8 ◽  
Author(s):  
Jing Wang ◽  
Na Xiong ◽  
Biao Li
Keyword(s):  
Time Reversal ◽  
Holm Equation ◽  
Novikov Equation ◽  
Delayed Time ◽  
Shifted Parity ◽  
Delayed Time Reversal

By requiring B=P^sT^dA and substituting u=A+B into the b-family equation and Novikov equation, we can obtain Alice-Bob peakon systems, where P^s and T^d are the arbitrary shifted parity transformation and delayed time reversal transformation, respectively. The nonlocal integrable Camassa-Holm equation and Degasperis-Procesi equation can be derived from the Alice-Bob b-family equations by choosing different parameters. Some new types of interesting solutions are solved including explicit one-peakons, two-peakons, and N-peakons solutions.

A general nonlocal variable coefficient KdV equation with shifted parity and delayed time reversal

2018 ◽  
Vol 94 (1) ◽  
pp. 693-702 ◽  
Author(s):  
Xiao-yan Tang ◽  
Shuai-jun Liu ◽  
Zu-feng Liang ◽  
Jian-yong Wang
Keyword(s):  
Time Reversal ◽  
Variable Coefficient ◽  
Kdv Equation ◽  
Delayed Time ◽  
Shifted Parity ◽  
Delayed Time Reversal

scholarly journals Symmetry Reduction, Exact Solutions, and Conservation Laws of the ZK-BBM Equation

2012 ◽  
Vol 2012 ◽  
pp. 1-9
Author(s):  
Wenbin Zhang ◽  
Jiangbo Zhou ◽  
Sunil Kumar
Keyword(s):  
Conservation Laws ◽  
Exact Solutions ◽  
Invariant Solution ◽  
Lie Symmetry ◽  
Similarity Reduction ◽  
Group Invariant ◽  
New Exact Solutions ◽  
Reduction Group ◽  
The Given ◽  
Bbm Equation

Employing the classical Lie method, we obtain the symmetries of the ZK-BBM equation. Applying the given Lie symmetry, we obtain the similarity reduction, group invariant solution, and new exact solutions. We also obtain the conservation laws of ZK-BBM equation of the corresponding Lie symmetry.

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.

New Exact Solutions for Two Nonlinear Equations

2006 ◽  
Vol 61 (1-2) ◽  
pp. 1-6 ◽  
Author(s):  
Zonghang Yang
Keyword(s):  
Wave Equation ◽  
Partial Differential Equations ◽  
Exact Solutions ◽  
Water Wave ◽  
Function Method ◽  
Nonlinear Partial Differential Equations ◽  
Soliton Solutions ◽  
New Exact Solutions ◽  
Complex Phenomena ◽  
Sivashinsky Equation

Nonlinear partial differential equations are widely used to describe complex phenomena in various fields of science, for example the Korteweg-de Vries-Kuramoto-Sivashinsky equation (KdV-KS equation) and the Ablowitz-Kaup-Newell-Segur shallow water wave equation (AKNS-SWW equation). To our knowledge the exact solutions for the first equation were still not obtained and the obtained exact solutions for the second were just N-soliton solutions. In this paper we present kinds of new exact solutions by using the extended tanh-function method.

Group Invariant Solutions and Conservation Laws of the Fornberg– Whitham Equation

2014 ◽  
Vol 69 (8-9) ◽  
pp. 489-496 ◽  
Author(s):  
Mir Sajjad Hashemi ◽  
Ali Haji-Badali ◽  
Parisa Vafadar
Keyword(s):  
Exact Solutions ◽  
Reduction Method ◽  
Lie Symmetry ◽  
First Integrals ◽  
Invariant Solutions ◽  
Analysis Method ◽  
Lie Symmetry Analysis ◽  
Group Invariant ◽  
Group Invariant Solutions ◽  
Whitham Equation

In this paper, we utilize the Lie symmetry analysis method to calculate new solutions for the Fornberg-Whitham equation (FWE). Applying a reduction method introduced by M. C. Nucci, exact solutions and first integrals of reduced ordinary differential equations (ODEs) are considered. Nonlinear self-adjointness of the FWE is proved and conserved vectors are computed

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