One-parameter Transformation Group Research Articles

Conserved vectors and solutions of the two-dimensional potential KP equation

Abstract This article investigates the potential Kadomtsev–Petviashvili (pKP) equation, which describes the evolution of small-amplitude nonlinear long waves with slow transverse coordinate dependence. For the first time, we employ Lie symmetry methods to calculate the Lie point symmetries of the equation, which are then utilized to derive exact solutions through symmetry reductions and with the help of Kudryashov’s method. The solutions obtained include exponential, hyperbolic, elliptic, and rational functions. Furthermore, we provide one-parameter group of transformations for the pKP equation. To gain a better understanding of the nature of each solution, we present 3D, 2D, and density plots. These obtained solutions, along with their associated physical characteristics, offer valuable insights into the propagation of small yet finite amplitude waves in shallow water.In addition, the pKP equation conserved vectors are derived by utilizing the multiplier method and the theorems by Noether and Ibragimov.

Nonequilibrium Geometric No-Arbitrage Principle and Asset Pricing Theorem

We find a novel and intimate correspondence in the present paper between the martingale and one-parameter transformation group and develop a nonequilibrium geometric no-arbitrage principle to a frictional financial market via this correspondence. Further, we achieve a fundamental pricing theorem via a geometric pricing transform (generator). Finally, we derive that the nonequilibrium geometric no-arbitrage is equivalent to NFLVR in a frictionless financial market. In addition, we apply the nonequilibrium geometric no-arbitrage condition to a frictional financial market. At the end of this paper, a numerical example confirms the effectiveness of the nonequilibrium geometric no-arbitrage condition.

Numerical validation of novel scaling laws for air entrainment in water

The Froude scaling laws have been used to model a wide range of water flows at reduced size for almost a century. In such Froude scale models, significant scale effects for air–water flows (e.g. hydraulic jumps or wave breaking) are typically observed. This study introduces novel scaling laws, excluding scale effects in the modelling of air–water flows. This is achieved by deriving the conditions under which the governing equations are self-similar. The one-parameter Lie group of point-scaling transformations is applied to the Reynolds-averaged Navier–Stokes equations, including surface tension effects. The scaling relationships between variables are derived for the flow variables, fluid properties and initial and boundary conditions. Numerical simulations are conducted to validate the novel scaling laws for (i) a dam break flow interacting with an obstacle and (ii) a vertical plunging water jet. Results for flow variables, void fraction and turbulent kinetic energy are shown to be self-similar at different scales, i.e. they collapse in dimensionless form. Moreover, these results are compared with those obtained using the traditional Froude scaling laws, showing significant scale effects. The novel scaling laws are a more universal and flexible alternative with a genuine potential to improve laboratory modelling of air–water flows.

Symmetry analysis of the generalized space and time fractional Korteweg–de Vries equation

In this paper, we studied the generalized space and time fractional Korteweg–de Vries (KdV) equation in the sense of the Riemann–Liouville fractional derivative. Initially, the symmetry of this considered equation through the symmetry analysis method was obtained. Next, a one-parameter Lie group of point transformation was yielded. Then, this considered fractional model can be translated into an ordinary differential equation of fractional order via the Erdélyi–Kober fractional differential operator and the Erdélyi–Kober fractional integral operator. Finally, with the help of the nonlinear self-adjointness, conservation laws of the generalized space and time fractional KdV equation can be found. This approach can provide us with a new scheme for studying space and time differential equations of fractional derivative.

An optimal system of group-invariant solutions and conserved quantities of a nonlinear fifth-order integrable equation

Abstract In this work, we perform Lie group analysis on a fifth-order integrable nonlinear partial differential equation, which was recently introduced in the literature and contains two dispersive terms. We determine a one-parameter group of transformations, an optimal system of group invariant solutions, and derive the corresponding analytic solutions. Topological kink, periodic and power series solutions are obtained. The existence of a variational principle for the underlying equation is proven using Helmholtz conditions and, thereafter, both local and nonlocal conserved quantities are obtained by utilising Noether’s theorem and a homotopy integral approach.

On integrability of the higher dimensional time fractional KdV-type equation

In this article, we investigated a nonlinear higher dimensional time fractional Korteweg–de Vries-type (KdV) equation. This considered equation is usually used to describe shallow water waves phenomena in physics. Here we found firstly the symmetry of the higher dimensional time fractional KdV-type equation in the sense of the Riemann–Liouville (RL) fractional derivative with the aid of the fractional Lie symmetry method. Then, the one-parameter group of Lie point symmetry transformation and some special solutions of this considered equation, were obtained. Next, the optimal system of one-dimensional Lie subalgebra of this considered equation, was constructed. Subsequently, on the basis of the multiple-parameter Erdélyi–Kober fractional differential operator (FDO) and Erdélyi–Kober fractional integral operator (FIO), the original equation can be reduced into the lower dimensional fractional differential equation (FDE). Furthermore, it can be translated further into more lower dimensional FDE with new symmetry from the reduced equation. Finally, conservation laws of this discussed equation are also found through a new conservation theorem. These results are favorable support for us to understand this dynamic model in a deeper level.

Boundary layer flow of a nanofluid past a horizontal flat plate in a Darcy porous medium: A Lie group approach

This research paper addresses the two-dimensional steady laminar incompressible free convective flow of a nanofluid past a horizontal plate saturated in the porous medium, where both the thermal and the mass convective boundary conditions are taken into consideration. Mathematical modeling via similarity transformations (which was developed using one-parameter continuous group of transformation) was applied to obtain a reduced mathematical model, which describes the problem. The solutions of the reduced system were obtained by a numerical method called the fourth- and fifth-order Runge–Kutta–Fehlberg method with the aid of computational software called Maple version 13. The resulting distributions of dimensionless temperature, velocity, and nanoparticle volume fraction are studied graphically to demonstrate the effect of pertinent parameters. Moreover, some of the new findings are shown in graphs. An excellent agreement was found after comparing the results with the previous literature, which assures the validity of the analysis. It is found that the flow is accelerated with an increase in thermal and mass convective parameters. Temperature and concentration are enhanced for rising values of (thermal and concentration) conjugate parameters.

Self-similarity in fate and transport of contaminants in groundwater

Similarity within non-linear geophysical processes under different space and time has been investigated in literature to understand and model the sophisticated nature of these processes. This study deals with the self-similarity of the governing processes of fate and transport of contaminants in groundwater, which include advection, dispersion, sorption, and degradation, either by chemical reaction or microbiological interaction. As such, self-similarity conditions of three-dimensional advective-dispersive-reactive transport (ADR) equation with various initial and boundary conditions are obtained by employing one-parameter Lie group of point scaling transformations. The strength of the proposed scaling approach lies in its ability to handle initial and boundary conditions of the process together with the governing equation. Then, utilizing reported conditions in Borden Field experiment domain, numerical simulations are performed to demonstrate examples of self-similarity when first-order degradation rate constant is zero and nonzero under three sorption conditions. The results of the numerical examples showed the effectiveness of the derived self-similarity conditions and the strength of the proposed approach in handling initial and boundary conditions of the governing transport process. It is possible to obtain both smaller and larger size self-similar (scaled-up or scaled-down) domains, which result in faster and slower transport processes, respectively. The obtained self-similarity criteria can offer further understanding of ADR transport phenomena and can provide spatial, temporal, and economical flexibility in constructing scaled models of field conditions governed by three-dimensional ADR transport process.

Integral generators of Archimedean n-copulas

ABSTRACT We introduce two algebraic structures induced by an Archimedean n-copula: a one-parameter group of transformations and a one-parameter local group of transformations. We then study the natural functional characteristics (integral generators) of these structures. We discuss differential equations joining the integral generators and the additive generators of copulas. We show that the algebraic structures can be recovered by using a certain parametric family of differential equations. Finally, we give some applications of integral generators, and we indicate potential applications of the algebraic and analytical results obtained.

Structure and thermodynamics of charged nonrotating black holes in higher dimensions

We analyze the structural and thermodynamic properties of $D$-dimensional ($D \geq 4$), asymptotically flat or Anti-de-Sitter, electrically charged black hole solutions, resulting from the minimal coupling of general nonlinear electrodynamics to General Relativity. This analysis deals with static spherically symmetric (elementary) configurations with spherical horizons. Our methods are based on the study of the behaviour (in vacuum and on the boundary of their domain of definition) of the Lagrangian density functions characterizing the nonlinear electrodynamic models in flat spacetime. These functions are constrained by some admissibility conditions endorsing the physical consistency of the corresponding theories, which are classified in several families, some of them supporting elementary solutions in flat space which are non topological solitons. This classification induces a similar one for the elementary black hole solutions of the associated gravitating nonlinear electrodynamics, whose geometrical structures are thoroughly explored. A consistent thermodynamic analysis can be developed for the subclass of families whose associated black hole solutions behave asymptotically as the Schwarzschild metric (in absence of a cosmological term). In these cases we obtain the behaviour of the main thermodynamic functions, as well as important finite relations among them. In particular, we find the general equation determining the set of extreme black holes for every model, and a general Smarr formula, valid for the set of elementary black hole solutions of such models. We also consider the one-parameter group of scale transformations, which are symmetries of the field equations of any nonlinear electrodynamics in flat spacetime.

On 3-Transitive Transformation Groups of the Lobachevskii Plane

In this paper, we consider three transformation groups of the Lobachevskii plane that are generated by the group of all motions and one-parameter transformation groups, which preserve an elliptic, a hyperbolic or a parabolic bundle of straight lines of this plane, respectively. It is proved that each of these groups acts 3-transitively on the Lobachevskii plane. The transformation groups and their generalizations can be applied an research of quasi-conformal mappings of the Lobachevskii space, in the special theory of relativity and in the fractal geometry.

Asymptotically Homogeneous Generalized Functions and Some of Their Applications

A brief description is given of generalized functions that are asymptotically homogeneous at the origin with respect to a multiplicative one-parameter transformation group such that the real parts of all eigenvalues of the infinitesimal matrix are positive. The generalized functions that are homogeneous with respect to such a group are described in full. Examples of the application of such functions in mathematical physics are given; in particular, they can be used to construct asymptotically homogeneous solutions of differential equations whose symbols are homogeneous polynomials with respect to such a group, as well as to study the singularities of holomorphic functions in tubular domains over cones.

Homogeneity of Infinite Dimensional Anti-Kaehler Isoparametric Submanifolds II

In this paper, we prove that, if a full irreducible infinite dimensional anti-Kaehler isoparametric submanifold of codimension greater than one has $J$-diagonalizable shape operators, then it is an orbit of the action of a Banach Lie group generated by one-parameter transformation groups induced by holomorphic Killing vector fields defined entirely on the ambient Hilbert space.

Lie symmetry analysis, Bäcklund transformations, and exact solutions of a (2 + 1)-dimensional Boiti-Leon-Pempinelli system

In this paper, the Lie symmetry analysis method is employed to investigate the Lie point symmetries and the one-parameter transformation groups of a (2 + 1)-dimensional Boiti-Leon-Pempinelli system. By using Ibragimov’s method, the optimal system of one-dimensional subalgebras of this system is constructed. Truncated Painlevé analysis is used for deriving the Bäcklund transformation. The method of constructing lump-type solutions of integrable equations by means of Bäcklund transformation is first presented. Meanwhile, the lump-type solutions of the (2 + 1)-dimensional Boiti-Leon-Pempinelli system are obtained. The lump-type wave is one kind of rogue wave. The fusion-type N-solitary wave solutions are also constructed. In addition, this system is integrable in terms of the consistent Riccati expansion method.

Group method analysis of mixed convection stagnation-point flow of non-Newtonian nanofluid over a vertical stretching surface

The group method analysis is applied to study the steady mixed convection stagnation-point flow of a non-Newtonian nanofluid towards a vertical stretching surface. The model utilized for the nanofluid incorporates the Brownian motion and thermophoresis effects. Applying the one-parameter transformation group which reduces the number of independent variables by one and thus, the system of governing partial differential equations has been converted to a set of nonlinear ordinary differential equations, and these equations are then computed numerically using the implicit finite-difference scheme. Comparison with previously published studies is executed and the results are found to be in excellent agreement. Results for the velocity, temperature, and the nanoparticle volume fraction profiles as well as the local skin-friction coefficient and local Nusselt number are presented in graphical and tabular forms, and discussed for different values of the governing parameters to show interesting features of the solutions.

The canonical group of transformations of a Weyl–Heisenberg frame; applications to Gaussian and Hermitian frames

We show, using a simple trick due to E. Fermi and rediscovered by Benenti and Strini, that to every Weyl–Heisenberg frame is associated in a canonical way a one-parameter group of symplectic transformations acting on the lattice of that frame without affecting the window. We study in detail the case of multidimensional Gaussian and Hermitian frames, and show that our method allows to create infinitely many new frames with same window staring from a single frame.

Noether’s theorem for fractional Birkhoffian systems of variable order

The Noether symmetries and conserved quantities of the Birkhoffian systems in terms of fractional derivatives of variable order are studied. Firstly, the Pfaff–Birkhoff–d’Alembert principle within fractional derivatives of variable order is obtained, and corresponding variable order fractional Birkhoff’s equations are deduced. Secondly, the invariance of the fractional Pfaff action of variable order is studied under the one-parameter group of infinitesimal transformations, and the definition of the variable order fractional conserved quantity is given. Finally, the Noether’s theorem for the fractional Birkhoffian system of variable order is established. At the end of this paper, an example is given to illustrate the application of the results.

Inverse problem of Mei symmetry for a general holonomic system

Inverse problems in dynamics are the basic problems in astronautics, rocket dynamics, and motion planning theory, etc. Mei symmetry is a kind of new symmetry where the dynamical function in differential equations of motion still satisfies the equation's primary form under infinitesimal transformations of the group. Mei symmetry and its inverse problem of dynamics for a general holonomic system in generalized coordinates are studied. Firstly, the direct problem of dynamics of the system is proposed and solved. Introducing a one-parameter infinitesimal transformation group with respect to time and coordinates, the infinitesimal generator vector and its first prolonged vector are obtained. Based on the discussion of the differential equations of motion for a general holonomic system determined by n generalized coordinates, their Lagrangian and non-potential generalized forces are made to have an infinitesimal transformation, the definition of Mei symmetry about differential equation of motion for the system is then provided. Ignoring the high-order terms in the infinitesimal transformation, the determining equation of Mei symmetry is given. With the aid of a structure equation which the gauge function satisfies, the system's corresponding conserved quantities are derived. Secondly, the inverse problem for the Mei symmetry of the system is studied. The formulation of the inverse problem of Mei symmetry is that we use the known conserved quantity to seek the corresponding Mei symmetry. The method is: considering a given integral as a Noether conserved quantity obtained by Mei symmetry, the generators of the infinitesimal transformations can be obtained by the inverse Noether theorem. Then the question whether the obtained generators are Mei symmetrical or not is verified by the determining equation, and the effect of generators' changes on the symmetries is discussed. It has been shown from the studies that the changes of the generators have no effect on the Noether and Lie symmetries, but have effects on the Mei symmetry. However, under certain conditions, while adjusting the gauge function, changes of generators can also have no effect on the Mei symmetry. In the end of the paper, an example for the system is provided to illustrate the application of the result.

Conformal invariance and conserved quantity of Mei symmetry for Appell equation in a holonomic system in relative motion

For a holonomic system in relative motion, the conformal invariance and the conserved quantity of Mei symmetry with Appell equations are investigated. First, by using the infinitesimal one-parameter transformation group and the infinitesimal generator vector, the definitions of Mei symmetry and the conformal invariance with Appell equations in a holonomic system in relative motion are given, and the determining equations of the conformal invariance of Mei symmetry for the system are derived. Relationship between the conformal invariance and Mei symmetry for the system is mainly studied. Then, by means of the structural equation which the gauge function satisfies, the expression of Mei conserved quantity deduced from Mei symmetry for the system is obtained. Finally, an example is given to illustrate the application of the result.

Noether's theorems of a fractional Birkhoffian system within Riemann—Liouville derivatives

The Noether symmetry and the conserved quantity of a fractional Birkhoffian system are studied within the Riemann—Liouville fractional derivatives. Firstly, the fractional Birkhoff's equations and the corresponding transversality conditions are given. Secondly, from special to general forms, Noether's theorems of a standard Birkhoffian system are given, which provide an approach and theoretical basis for the further research on the Noether symmetry of the fractional Birkhoffian system. Thirdly, the invariances of the fractional Pfaffian action under a special one-parameter group of infinitesimal transformations without transforming the time and a general one-parameter group of infinitesimal transformations with transforming the time are studied, respectively, and the corresponding Noether's theorems are established. Finally, an example is given to illustrate the application of the results.