Optimal System Of One-dimensional Subalgebras Research Articles

One class of solutions to the equations of ideal plasticity

Much attention is given to the study and solution of nonlinear differential equations in the modern mathematical literature. Despite this, there are not many methods for researching and solving such equations. These are point and contact transformations of equations, various methods of separating variables, the method of differential connections, the search for various symmetries and their use to construct solutions, as well as conservation laws. The paper considers a nonlinear differential equation describing the plastic flow of a prismatic rod. A group of point symmetries is found for this equation. The optimal system of one-dimensional subalgebras is calculated. Conservation laws corresponding to Noetherian symmetries are given, and it is also shown that there are infinitely many non-Noetherian conservation laws. Several new invariant solutions of rank one, i. e. depending on one independent variable, are constructed. It is shown how classes of new solutions can be constructed from two exact solutions, passing to a linear equation. Thus, in this short article, almost all methods of modern research of nonlinear differential equations are involved.

Novel dynamical group-invariant solutions and conserved vectors of the Gilson–Pickering equation with applications in plasma physics

This work delves into the analysis of the Gilson–Pickering equation which governs the waves propagation in plasma physics by invoking Lie symmetry analysis. We commence by identifying the Lie point symmetries associated with the equation. These symmetries are then leveraged on to compute the commutator table and subsequently the adjoint representation, ultimately leading to the establishment of an optimal system of one-dimensional subalgebras. Each subalgebra within this system is subsequently utilized to perform symmetry reductions. Through these reductions, various forms of nonlinear ordinary differential equations are obtained, which are subsequently solved using the power series method and Kudryashov’s technique. The resulting solutions are given in terms of hyperbolic functions. To gain deeper insights into the behaviour of these solutions, three-dimensional and two-dimensional plots are presented. Furthermore, applying the Ibragimov’s theorem allows us to derive conserved vectors associated with the Gilson–Pickering equation.

Application of conserved quantities using the formal Lagrangian of a nonlinear integro partial differential equation through optimal system of one-dimensional subalgebras in physics and engineering

Abstract This research article analytically investigates a soliton equation of high dimensions, particularly with applications, and precisely in the fields of physical sciences and engineering. The soliton equation of high dimensions, particularly with applications, and precisely in the fields of physical sciences along with engineering, is examined with a view to securing various pertinent results of interest. For the first time, the conserved currents of an integrodifferential equation (especially those of higher dimensions) are calculated using a detailed optimal system of one-dimensional subalgebras. Infinitesimal generators of diverse structures ascribed to Lie point symmetries of the understudy model are first calculated via Lie group analysis technique. Additionally, we construct various commutations along Lie-adjoint representation tables connected to the nine-dimensional Lie algebra achieved. Further to that, detailed and comprehensive computation of the optimal system of one-dimensional subalgebras linked to the algebra is also unveiled for the under-investigated model. This, in consequence, engenders the calculation of abundant conserved currents for the soliton equation through Ibragimov’s conserved vector theorem by utilizing its formal Lagrangian. Later, the applications of our results are highlighted.

Lie symmetry analysis, conservation laws and exact solutions for variable-coefficients (2+1)-dimensional dissipative long-wave system

The Lie symmetry analysis, optimal system, exact solutions and conservation laws of the (2+1)-dimensional variable-coefficients dissipative long-wave (vcDLW) system are introduced in this paper. Lie symmetry analysis method is used to obtain the vector field firstly. And the Olver method is used to obtain the optimal system of one-dimensional subalgebras of the equations. Based on the optimal system, the equations are similarly reduced. Some representative equations are selected from the equations obtained after similarly reduction, and then the expansion method is applied to these equations, so that some new exact solutions of the (2+1)-dimensional vcDLW equations can be obtained. Finally, we study the conservation laws of the system.

An invariance analysis of the Vakhnenko–Parkes Equation

Lie symmetry analysis of differential equations is an effective technique to compute exact solutions of a differential equation, to reduce number of independent variables or to reduce the order and nonlinearity of the equation. The present article focuses on the symmetry analysis of Vakhnenko–Parkes Equation (VP Equation). At first, Lie point symmetries of the VP Equation that permit an optimal system of one-dimensional subalgebras are discussed and then using the commutator table we construct the discrete symmetries of the VP Equation using the modified Hydon’s method. Also, we compute the exact solutions using symmetry transformations.

Conservation laws and symmetry analysis of a generalized Drinfeld-Sokolov system

<abstract><p>The generalized Drinfeld-Sokolov system is a widely-used model that describes wave phenomena in various contexts. Many properties of this system, such as Hamiltonian formulations and integrability, have been extensively studied and exact solutions have been derived for specific cases. In this paper we applied the direct method of multipliers to obtain all low-order local conservation laws of the system. These conservation laws correspond to physical quantities that remain constant over time, such as energy and momentum, and we provided a physical interpretation for each of them. Additionally, we investigated the Lie point symmetries and first-order symmetries of the system. Through the point symmetries and constructing the optimal systems of one-dimensional subalgebras, we were able to reduce the system of partial differential equations to ordinary differential systems and obtain new solutions for the system.</p></abstract>

Preliminary Group Classification of Nonlinear Wave Equation utt + ut = f(x,ux)uxx + g(x,ux)

The paper discusses the non-linear wave equations whose coefficients are dependent on first order spatial derivatives. We construct the principal Lie algebra, the equivalence Lie algebra, and the extensions by one of the principal Lie algebra. We further construct the optimal system of one-dimensional subalgebras for first three extended five-dimensional Lie algebras. These are finally used to determine invariant solutions of some examples.

Lie symmetry analysis, optimal system and exact solutions of variable-coefficients Sakovich equation

In this paper, the Sakovich equation is extended for the first time to a new equation with variable-coefficients on the time variable. The infinitesimal generators are obtained by performing a Lie symmetry analysis of the equation. Following this, the optimal system of one-dimensional subalgebras of the equation is analyzed. The (2+1)-dimensional variable-coefficients Sakovich equation is then reduced to (1+1)-dimensional partial differential equations by similarity reductions. The reduced partial differential equations are simplified to ordinary differential equations by the traveling wave transform. The new periodic wave solution and two-soliton interaction solution are obtained, and some exact solutions are obtained using the extended tanh-function method and the extended sech-function method. Finally, different types of soliton solutions are obtained by assigning different functions to the coefficient functions, and 3-dimensional figures are used to visualize the structural features of the soliton solutions.

Optimal system, invariant solutions and conservation laws of the hyperbolic geometry flow with time-dependent dissipation

Based on the classical Lie symmetry method, the hyperbolic geometry flow with time-dependent dissipation is discussed. In addition, we construct the optimal system of one-dimensional subalgebras, and invariant solutions of two different dissipation equations are obtained, which include traveling wave solutions. Finally, the conservation laws are obtained for geometry flow with different two dissipation term.

Symmetry group analysis and conservation laws of the potential modified KdV equation using the scaling method

In this paper, using the Lie symmetry method, we obtain the Lie symmetry group of the potential modified Korteweg–de Vries (potential mKdV) equation, and we construct the optimal system of one-dimensional subalgebras for the Lie symmetry algebra of this equation. Furthermore, similarity reductions related to the infinitesimal generators of the Lie symmetry group are obtained. Also, we construct new conservation laws for this equation by the scaling method. This method is algorithmic and based on the linear algebra tools and calculus of variations. The density of the conservation laws is constructed using the Euler operator and scaling symmetry of the equation and also the associated flux is calculated by the homotopy operator. This density-flux pair renders a conservation law for the potential mKdV equation. We specify the density-flux pairs of conservation laws in ranks 4, 8, 12, 16 and 20 for the potential mKdV equation.

Lie symmetry analysis, exact solutions, conservation laws of variable-coefficients Calogero–Bogoyavlenskii–Schiff equation

In this paper, a [Formula: see text]-dimensional variable-coefficients Calogero–Bogoyavlenskii–Schiff (vcCBS) equation is studied. The infinitesimal generators and symmetry groups are obtained by using the Lie symmetry analysis on vcCBS. The optimal system of one-dimensional subalgebras of vcCBS is computed for determining the group-invariant solutions. On this basis, the vcCBS is reduced to two-dimensional partial differential equations (PDEs) by similarity reductions. Furthermore, the reduced PDEs are solved to obtain the two-soliton interaction solution, the soliton-kink interaction solution and some other exact solutions by the [Formula: see text]-expansion method. Moreover, it is shown that vcCBS is nonlinearly self-adjoint and then its conservation laws are calculated.

Lie symmetry analysis of a pseudoparabolic PDE: Power law in diffusion coefficient with constant viscosity

We perform symmetry analysis of a partial differential equation (PDE) of pseudoparabolic form which models solvent uptake in polymeric solids; a special case of power law for diffusion coefficient and constant viscosity is considered. Optimal systems of one-dimensional subalgebras for various model parameters are employed for symmetry reductions and construction of invariant solutions. • A phenomenon of case II diffusion in glassy polymers is considered. • A model equation is an implicit pseudoparabolic partial differential equation which incorporates the diffusion coefficient and viscosity of the polymer. • The model equation is studied from Lie symmetry analysis view point for a special case of power law in diffusion coefficient and constant viscosity. • Some exact solutions are obtained and presented graphically.

Lie Symmetry Analysis, Exact Solutions, and Conservation Laws of Variable-Coefficients Boiti-Leon-Pempinelli Equation

In this article, we study the generalized ( 2 + 1 )-dimensional variable-coefficients Boiti-Leon-Pempinelli (vcBLP) equation. Using Lie’s invariance infinitesimal criterion, equivalence transformations and differential invariants are derived. Applying differential invariants to construct an explicit transformation that makes vcBLP transform to the constant coefficient form, then transform to the well-known Burgers equation. The infinitesimal generators of vcBLP are obtained using the Lie group method; then, the optimal system of one-dimensional subalgebras is determined. According to the optimal system, the ( 1 + 1 )-dimensional reduced partial differential equations (PDEs) are obtained by similarity reductions. Through G ′ / G -expansion method leads to exact solutions of vcBLP and plots the corresponding 3-dimensional figures. Subsequently, the conservation laws of vcBLP are determined using the multiplier method.

Symmetry Analysis, Exact Solutions and Conservation Laws of a Benjamin–Bona–Mahony–Burgers Equation in 2+1-Dimensions

The Benjamin–Bona–Mahony equation describes the unidirectional propagation of small-amplitude long waves on the surface of water in a channel. In this paper, we consider a family of generalized Benjamin–Bona–Mahony–Burgers equations depending on three arbitrary constants and an arbitrary function G(u). We study this family from the standpoint of the theory of symmetry reductions of partial differential equations. Firstly, we obtain the Lie point symmetries admitted by the considered family. Moreover, taking into account the admitted point symmetries, we perform symmetry reductions. In particular, for G′(u)≠0, we construct an optimal system of one-dimensional subalgebras for each maximal Lie algebra and deduce the corresponding (1+1)-dimensional nonlinear third-order partial differential equations. Then, we apply Kudryashov’s method to look for exact solutions of the nonlinear differential equation. We also determine line soliton solutions of the family of equations in a particular case. Lastly, through the multipliers method, we have constructed low-order conservation laws admitted by the family of equations.

Conservation laws, classical symmetries and exact solutions of a (1 + 1)-dimensional fifth-order integrable equation

In this paper, we present a study of a fifth-order nonlinear partial differential equation, which was recently introduced in the literature. This equation can be used as a model for bidirectional water waves propagating in a shallow medium. Using elements of an optimal system of one-dimensional subalgebras, we perform similarity reductions culminating in analytic solutions. Rational, hyperbolic, power series and elliptic solutions are obtained. Furthermore, by using the multiple exponential function method we obtain one and two soliton solutions. Finally, local and low-order conserved quantities are derived by enlisting the multiplier approach.

A study of the generalized nonlinear advection-diffusion equation arising in engineering sciences

In this work, we examine a nonlinear partial differential equation of fluid mechanics, namely, the generalized nonlinear advection–diffusion equation, which portrays the motion of buoyancy driven plume in a bent-on porous medium. Firstly, we classify all (point) symmetries of the equation, which prompt three cases of n. Next, for each case, we construct an optimal system of one-dimensional subalgebras and use them to perform symmetry reductions and symmetry invariant solutions. In a bid to explain the physical significance of some invariant solutions secured, we present a graphic display of some solutions in 3D, 2D as well as density plots via the exploitation of numerical simulations. Besides, we categorically state here that the results obtained in this study are new when compared with the outcomes previously achieved by Loubens et al., 2011 Quart. Appl. Math. 69 389–401. Interestingly, kink shape soliton, dark soliton, singular soliton together with exponential function solution wave profiles are displayed to make this work more valuable. Furthermore, we determine the conserved vectors in two different ways: engaging the general multiplier approach and Ibragimov’s conservation law theorem. Finally, we provide the physical meaning of these conservation laws.

Invariant Solutions and Conservation Laws of the Variable-Coefficient Heisenberg Ferromagnetic Spin Chain Equation

The variable-coefficient Heisenberg ferromagnetic spin chain (vcHFSC) equation is investigated using the Lie group method. The infinitesimal generators and Lie point symmetries are reported. Four types of similarity reductions are acquired by virtue of the optimal system of one-dimensional subalgebras. Several invariant solutions are derived, including trigonometric and hyperbolic function solutions. Furthermore, conservation laws for the vcHFSC equation are obtained with the help of Lagrangian and nonlinear self-adjointness.

On group analysis of the time fractional extended (2+1)-dimensional Zakharov–Kuznetsov equation in quantum magneto-plasmas

In this article, the time fractional extended (2+1)-dimensional Zakharov–Kuznetsov (Z–K) equation in quantum magneto-plasmas, is executed. First of all, the symmetry of this considered equation by using group analysis approach with the sense of Riemann–Liouville (R–L) fractional derivative, is obtained. Then, the symmetry of the above yielded, the optimal system of one-dimensional subalgebras for this equation is also found. Subsequently, the original equation can be reduced into (1+1)-dimensional fractional differential equation with the adding of extended Erdélyi–Kober fractional differential operator. Further, the one parameter group, invariant solutions and non-invariant solutions are constructed. Finally, the conservation laws are also shown with a new conservation theorem. We believe that these beautiful results can help us to discover more evolutionary mechanisms of the considered equation.

Lie symmetries, optimal system and dynamics of exact solutions of (2+1)-dimensional KP-BBM equation

The present research is devoted to carry out Lie group classification and optimal system of one-dimensional subalgebras of KP–BBM equation. The equation describes bidirectional small amplitude and weakly dispersive long waves in nonlinear dispersive systems. The infinitesimal generators for the governing equation have been derived under invariance property of Lie groups. Thereafter, Lie symmetry analysis is used to derive commutative relations, invariant functions and optimal syatem. The symmetry reductions of KP–BBM equation are derived on basis of optimal system. Meanwhile, the twice reductions transform the KP–BBM equation into overdetermined ODEs, which lead to the exact solutions. In order to analyze the behavior of phenomena physically, the obtained solutions are extended with numerical simulation. Thus, doubly soliton, elastic multisoliton, compacton, bright and dark soliton profiles of solutions are presented to make this research physically meaningful.

Optimal System and Invariant Solutions of a New AKNS Equation with Time-Dependent Coefficients

The Lie point symmetries are reported by performing the Lie symmetry analysis to the Ablowitz-Kaup-Newell-Suger (AKNS) equation with time-dependent coefficients. In addition, the optimal system of one-dimensional subalgebras is constructed. Based on this optimal system, several categories of similarity reduction and some new invariant solutions for the equation are obtained, which include power series solutions and travelling and non-traveling wave solutions.