In this paper, a generalized (2+1)-dimensional Hirota–Satsuma–Ito (GHSI) equation is investigated using Lie symmetry approach. Infinitesimal generators and symmetry groups of this equation are presented, and the optimal system is given with adjoint representation. Based on the optimal system, some symmetry reductions are performed and some similarity solutions are provided, including soliton solutions and periodic solutions. With Lagrangian, it is shown that the GHSI equation is nonlinearly self-adjoint. By means of the Lie point symmetries and nonlinear self-adjointness, the conservation laws are constructed. Furthermore, some physically meaningful solutions are illustrated graphically with suitable choices of parameters.
In this paper we study the fourth-order three-dimensional generalized potential Yu-Toda-Sasa-Fukuyama (gpYTSF) equation by first computing its Lie point symmetries and then performing symmetry reductions. The resulting ordinary differential equations are then solved using direct integration, and exact solutions of gpYTSF equation are obtained. The obtained group invariant solutions include the solution in terms of incomplete elliptic integral. Furthermore, conservation laws for the gpYTSF equation are derived using both the multiplier and Noether’s methods. The multiplier method provides eight conservation laws, while the Noether’s theorem supplies seven conservation laws. These conservation laws include the conservation of energy and mass.
In this article, we adopt two kinds of loop algebras corresponding to the Lie algebra B(0,1) to introduce two line spectral problems with different numbers of even and odd superfunctions. Through generalizing the time evolution λt to a polynomial of λ, two isospectral-nonisospectral super integrable hierarchies are derived in terms of Tu scheme and zero-curvature equation. Among them, the first super integrable hierarchy is further reduced to generalized Fokker–Plank equation and special bond pricing equation, as well as an explicit super integrable system under the choice of specific parameters. More specifically, a super integrable coupled equation is derived and the corresponding integrable properties are discussed, including the Lie point symmetries and one-parameter Lie symmetry groups as well as group-invariant solutions associated with characteristic equation.
This paper studies a non-linear viscoelastic wave equation, with non-linear damping and source terms, from the point of view of the Lie groups theory. Firstly, we apply Lie’s symmetries method to the partial differential equation to classify the Lie point symmetries. Afterwards, we reduce the partial differential equation to some ordinary differential equations, by using the symmetries. Therefore, new analytical solutions are found from the ordinary differential equations. Finally, we derive low-order conservation laws, depending on the form of the damping and source terms, and discuss their physical meaning.
In general, due to the nature of the Lie group theory, symmetry analysis is applied to single equations rather than boundary value problems. In this paper boundary value problems for the sine-Gordon equations under the group of Lie point symmetries are obtained in both differential and difference forms. The invariance conditions for the boundary value problems and their solutions are obtained. The invariant discretization of the difference problem corresponding to the boundary value problem for sine-Gordon equation is studied. In the differential case an unbounded domain is considered and in the difference case a lattice with points lying in the plane and stretching in all directions with no boundaries is considered.
We investigate the relation of the Lie point symmetries for the geodesic equations with the collineations of decomposable spacetimes. We review previous results in the literature on the Lie point symmetries of the geodesic equations and we follow a previous proposed geometric construction approach for the symmetries of differential equations. In this study, we prove that the projective collineations of a n+1-dimensional decomposable Riemannian space are the Lie point symmetries for geodesic equations of the n-dimensional subspace. We demonstrate the application of our results with the presentation of applications.
Lie Point symmetries and Euler’s formula for solving second order ordinary linear differential equations are used to determine symmetries for the one-dimensional Black- Scholes equation. One symmetry is utilized to determine an invariant solutions
Symmetry Solutions and Conservation Laws for the 3D Generalized Potential Yu-Toda-Sasa-Fukuyama Equation of Mathematical Physics
