Lie Symmetry Analysis and Conservation Laws for the (2 + 1)-Dimensional Dispersionless B-Type Kadomtsev–Petviashvili Equation

Zhao, Qiulan; Wang, Huanjin; Li, Xinyue; Li, Chuanzhong

doi:10.1007/s44198-022-00073-6

Cited by 5 publications

(2 citation statements)

References 29 publications

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“…Lie symmetries have been widely employed on dimension and order reductions for PDEs. They have effectively derived explicit invariant solutions, such as traveling wave, fundamental, and soliton solutions, which could display the structure of the singularities or describe the asymptotic behavior of a general solution [18][19][20]. Since each Lie symmetry of a PDE could result in the automorphism of its Lie algebra, all discrete symmetries could be derived from the known Lie symmetries [21].…”

Section: Introductionmentioning

confidence: 99%

Novel symmetric structures and explicit solutions to a coupled Hunter-Saxton equation

Zhao

Wang

2023

Phys. Scr.

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In the current study, novel symmetric structures to a coupled Hunter-Saxton equation are synthetically investigated. These novel symmetric structures include Lie symmetries, discrete symmetries, nonlocally related systems, and $ \mu $-symmetries. Lie symmetries and $ \mu $-symmetries are then used to derive explicit invariant solutions. Based on the established optimal system, the coupled Hunter-Saxton equation can be reduced to rich ordinary differential equations by the Lie group transformation. Its group invariant solutions are thus obtained. Discrete symmetries to the coupled Hunter-Saxton equation are constructed utilizing Lie symmetries, which can help calculate new solutions from known explicit solutions. Moreover, nonlocally related systems of the coupled Hunter-Saxton equation are completed, which contain potential systems and inverse potential systems based on conservation laws and Lie symmetries, respectively.\@ Furthermore, without using the group theory, more plentiful similarity reductions and similarity solutions to the coupled Hunter-Saxton equation are produced by employing the direct reduction method.\@ Another class of symmetric structures to the coupled Hunter-Saxton equation explored in this paper are $ \mu $-symmetries, which are given by matching an integrable and horizontal one-form $ \mu =\varLambda_{x}dx+\varLambda_{t}dt $ for Lie symmetries. Hence, $ \mu $-reductions, explicit solutions and $ \mu $-conservation laws can be determined by $ \mu $-symmetries. In addition, polynomial solutions are researched by considering the linear invariant subspaces admitted by the coupled Hunter-Saxton equation. Several explicit invariant solutions are described by graphs ultimately.

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Section: Introductionmentioning

confidence: 99%

Novel symmetric structures and explicit solutions to a coupled Hunter-Saxton equation

Zhao

Wang

2023

Phys. Scr.

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“…However, constructing nonlocally related PDE systems for multidimensional PDEs is more challenging and meaningful since these PDEs are more widespread in mathematics and physics [21][22][23]. Especially for threedimensional cases, it is necessary to introduce three ( m 2 −m 2 ) potentials to write a three-dimensional PDE as an equivalent divergence-type CLs-based potential system (Poincaré's lemma).…”

Section: Introductionmentioning

confidence: 99%

The extended research for constructing three-dimensional nonlocally related partial differential equation systems

Wang

Zhao

2023

Preprint

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The coordinate-independent analysis (qualitative, numerical, perturbation, etc.) on a given partial differential equation (PDE) may be effectively facilitated if its nonlocally related PDE systems can be constructed. The present research extendes their construction in three-dimensional situations, including the reduced potential system (RPS) based on the divergence-type conservation law (CL) and the inverse potential system (IPS) based on the differential invariant (DI). For the RPS, the breakthrough is to prove that the potential system with the algebraic gauge constraint can further reduce the potential without weakening the solution space. The RPS with a more compact structure not only simplifies the computation of symmetric properties (equivalent transformations, nonlocal symmetries), but also promotes to produce more valuable results (nonlocal CLs and the expansion of the nonlocal CLs based-RPS). For the IPS, the more critical breakthrough is to generalize the previous theorems from two-dimensional to three-dimensional. It consists of nonlocal symmetries’ identification by the IPS, and the IPS’s expansion by the solvable Lie algebra. As the application, this research is demonstrated with a typical three-dimensional PDE with the power law nonlinearity and multiple parameters. Once the parameters are determined, our analysis corresponds to several equations describing turbulence, shock waves, and weakly diffracted waves. Concretely, its RPSs, IPSs, their expansions and associatied symmetric properties are derived. Consequently, these generated systems are nonlocally related to the demonstrated equation, constituting an expanded tree of nonlocally related PDE systems. Moreover, this research can aslo be applied to other three-dimensional PDEs that admit CLs or DIs.

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Symmetric structures and dynamic analysis of a (2+1)-dimensional generalized Benny-Luke equation

Sun,

Zhao,

2024

Phys. Scr.

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We study the symmetric structures and dynamic analysis of a (2 + 1)-dimensional generalized Benny-Luke (GBL) equation based on the Lie symmetry method, the GBL equation is an important non-integrable model of water waves. Speciﬁcally, we construct multiple exact solutions of the GBL equation and obtain its nonlocally related systems. Firstly, the Lie point symmetries and conservation laws of the GBL equation are computed, and then we get the reduced ordinary diﬀerential equation from one of the conservation laws. Multiple methods, for example, the dynamic system method, the power series method, the homogeneous balancing method and generalized variable separation method, are used to solve the ordinary diﬀerential equation and abundant exact solutions of the GBL equation are got. Finally, we extend these exact solutions by discrete symmetries, and give three-dimensional graphs of partial exact solutions. In addition, we construct the nonlocally related systems, which contains the potential systems from the conservation laws and an inverse system from a Lie point symmetry of the GBL equation. These ﬁndings reveal the dynamical behavior behind the GBL equation and broaden the range of nonlinear water wave model solutions.

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Lie Symmetry Analysis and Conservation Laws for the (2 + 1)-Dimensional Dispersionless B-Type Kadomtsev–Petviashvili Equation

Cited by 5 publications

References 29 publications

Novel symmetric structures and explicit solutions to a coupled Hunter-Saxton equation

Novel symmetric structures and explicit solutions to a coupled Hunter-Saxton equation

The extended research for constructing three-dimensional nonlocally related partial differential equation systems

Symmetric structures and dynamic analysis of a (2+1)-dimensional generalized Benny-Luke equation

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