Narrow Results

We study the (2+1)-dimensional Schwarzian Korteweg–de Vries equation (SKdV). The explored solutions describe new Lump soliton colliding with visible soliton, an interaction between multi-soliton waves with one soliton, multi peaks of waves moving in a curved path, two hyperbolic waves moving together without interaction and some of periodic waves. We examine the commutative product between multi unknown Lie infinitesimals for the (2+1)-dimensional (SKdV) equation, and this study result in some new Lie vectors. The commutative product generates a system of nonlinear ODEs which had been solved manually. Through two stages of Lie symmetry reduction, SKdV equation is reduced to non-solvable nonlinear ODEs using various combinations of optimal Lie vectors. Using the Riccati–Bernoulli sub-ODE and Integration methods, we investigate new analytical solutions for these ODEs. Back substituting for the original variables generates new solutions for SKdV. Some selected solutions are illustrated through three-dimensional plots.

For the inflaton field, we determine a new exact solution by using the Lie symmetry analysis. Specifically, we construct a second-order differential master equation for arbitrary scalar field potential by assuming that the spectral indices for the density perturbations $n_s$ and the scalar-to-tensor ratio $r$ are related as $n_s-1=h(r)$. Function $h(r)$ is classified according to the admitted Lie symmetries for the master equation. The possible admitted Lie symmetries form the $A_2$, $A_{3,2}$, $A_{3,3}$ and $sl(3,R)$ Lie algebras. The new inflationary solution is recovered by the Lie symmetries of the $A_{3,3}$ algebra. Scalar field potential is derived explicitly, while we compare the resulting spectral indices with the observations.

We present a complete algebraic classification of the Lie symmetries for generalized Zoomeron equations. For the generalized $1+1$ and $2+1$ Zoomeron equations we solve the Lie symmetry conditions in order to constrain the free functions of the equations. We find that the differential equations of our consideration admit the same number of Lie symmetries with the non-generalized equations. The admitted Lie symmetries form the Lie algebras $2A_1$, $A_{3,3}$ for the $1+1$ generalized Zoomeron equation, and the $A_{4,5}^{ab}$, $3A_1{\otimes }_{s}2A_1$ in the case of the $2+1$ generalized Zoomeron equation. The one-dimensional optimal system is constructed for the two equations and similarity solutions are derived. The similarity transformation led to the derivation of kink solutions. Indeed, the similarity exact solutions determined in this work are asymptotic solutions near the singular behavior of the kink behavior.

In this paper, we perform a detailed group classification for a generalized convection-reaction-diffusion equation with three unknown functions. Specifically, we determine all the functional forms for the unknown functions where the given equation admits nontrivial Lie point symmetries. The classification problem provides us with eight families of equations summarized in four categories. The admitted Lie symmetries form the four Lie algebras $2A_1$, $A_{4,4}$, $A_{2,1}\oplus A_1$ and $A_{2,1}\oplus A_{2,1}$. For the four families of the classification problem we calculate the one-dimensional optimal system and we derive all the similarity transformations which reduce the partial differential equation into an ordinary differential equation. Applications of the similarity transformations are presented while exact solutions are derived.

We use group theoretic methods to obtain the extended Lie point symmetries of the quantum dynamics of a scalar particle probing the near horizon structure of a black hole. Symmetries of the classical equations of motion for a charged particle in the field of an inverse square potential and a monopole, in the presence of certain model magnetic fields and potentials are also studied. Our analysis gives the generators and Lie algebras generating the inherent symmetries.

We present a family of networks whose local interconnection topologies are generated by the root vectors of a semi-simple complex Lie algebra. Cartan classification theorem of those algebras ensures those families of interconnection topologies to be exhaustive. The global arrangement of the network is defined in terms of integer or half-integer weight lattices. The mesh or torus topologies that network millions of processing cores, such as those in the IBM BlueGene series, are the simplest member of that category. The symmetries of the root systems of an algebra, manifested by their Weyl group, lends great convenience for the design and analysis of hardware architecture, algorithms and programs.

In this work, we consider a damped equation with a time-independent source term. We derive the classical Lie symmetries admitted by the equation as well as the reduced ordinary differential equations. We also present some exact solutions. Conservation laws for this equation are constructed by using the multiplier method.

In the present paper, we perform Lie and Noether symmetries of the generalized Klein–Gordon–Fock equation. It is shown that the principal Lie algebra, which is one-dimensional, has several possible extensions. It is further shown that several cases arise for which Noether symmetries exist. Exact solutions for some cases are also obtained from the invariant solutions of the investigated equation.

In this paper, the Oskolkov–Benjamin–Bona–Mahony–Burgers (OBBMB) equation has been investigated by Lie symmetry analysis. Lie group analysis method is implemented to derive the vector fields and symmetry reductions. The OBBMB equation has been reduced into nonlinear ordinary differential equation (ODE) by exploiting symmetry reduction method. Using the similarity reduction equation, the exact solutions are obtained by extended $(G^{\prime }/G)$-method. Finally, the new conservation theorem proposed by Ibragimov has been utilized to construct the conservation laws of the aforesaid equation.

The dynamics of nonlinear reaction–diffusion systems is dominated by the onset of patterns, and Fisher equation is considered to be a prototype of such diffusive equations. Here we investigate the integrability properties of a generalized Fisher equation in both (1+1) and (2+1) dimensions. A Painlevé singularity structure analysis singles out a special case (m=2) as integrable. More interestingly, a Bäcklund transformation is shown to give rise to a linearizing transformation for the integrable case. A Lie symmetry analysis again separates out the same m=2 case as the integrable one and hence we report several physically interesting solutions via similarity reductions. Thus we give a group theoretical interpretation for the system under study. Explicit and numerical solutions for specific cases of nonintegrable systems are also given. In particular, the system is found to exhibit different types of traveling wave solutions and patterns, static structures and localized structures. Besides the Lie symmetry analysis, nonclassical and generalized conditional symmetry analysis are also carried out.

Thin films arise in a variety of contexts. They can be between two solid surfaces in the form of a lubricant or with one surface solid and the other free as in water or oil on a road. When the surface tension is a dominant feature of the physics behind the motion, the governing evolution of partial differential equation is nonlinear and of the fourth order in the spatial derivatives. A particular form proposed by King [2001] invites generalization to create a class of equations of higher order. We examine some of the properties of this class in terms of symmetry and singularity and see that there are some open questions which require investigation using different methodologies.

We show that the recent results of [S. Dutta and S. Chakraborty, Int. J. Mod. Phys. D25 (2016) 1650051] on the application of Lie/Noether symmetries in scalar field cosmology are well-known in the literature while the problem could have been solved easily under a coordinate transformation. That follows from the property, that the admitted group of invariant transformations of dynamical system is independent on the coordinate system.

Asian options incorporate the average stock price in the terminal payoff. Examination of the Asian option partial differential equation (PDE) has resulted in many equations of reduced order that in general can be mapped into each other, although this is not always shown. In the literature these reductions and mappings are typically acquired via inspection or ad hoc methods. In this paper, we evaluate the classical Lie point symmetries of the Asian option PDE. We subsequently use these symmetries with Lie's systematic and algorithmic methods to show that one can obtain the same aforementioned results. In fact we find a familiar analytical solution in terms of a Laplace transform. Thus, when coupled with their methodic virtues, the Lie techniques reduce the amount of intuition usually required when working with differential equations in finance.

We determine the Lie point symmetries of the Schrödinger and the Klein–Gordon equations in a general Riemannian space. It is shown that these symmetries are related with the homothetic and the conformal algebra of the metric of the space, respectively. We consider the kinematic metric defined by the classical Lagrangian and show how the Lie point symmetries of the Schrödinger equation and the Klein–Gordon equation are related with the Noether point symmetries of this Lagrangian. The general results are applied to two practical problems: (a) The classification of all two- and three-dimensional potentials in a Euclidean space for which the Schrödinger equation and the Klein–Gordon equation admit Lie point symmetries; and (b) The application of Lie point symmetries of the Klein–Gordon equation in the exterior Schwarzschild spacetime and the determination of the metric by means of conformally related Lagrangians.

In this work we perform the symmetry classification of the Klein–Gordon equation in Bianchi I spacetime. We apply a geometric method which relates the Lie symmetries of the Klein–Gordon equation with the conformal algebra of the underlying geometry. Furthermore, we prove that the Lie symmetries which follow from the conformal algebra are also Noether symmetries for the Klein–Gordon equation. We use these results in order to determine all the potentials in which the Klein–Gordon admits Lie and Noether symmetries. Due to the large number of cases and for easy reference the results are presented in the form of tables. For some of the potentials we use the Lie admitted symmetries to determine the corresponding invariant solution of the Klein–Gordon equation. Finally, we show that the results also solve the problem of classification of Lie/Noether point symmetries of the wave equation in Bianchi I spacetime and can be used for the determination of invariant solutions of the wave equation.

Two essential methods, the symmetry analysis and the singularity analysis, for the study of the integrability of nonlinear ordinary differential equations is the purpose of this work. The main similarities and the differences of these two different methods are discussed.

In this paper, for the study of integrability, symmetry analysis, group invariant solutions and conservation laws, the Mikhailov–Novikov–Wang equation is considered. Firstly, Painlevé analysis is being employed to study the integrability properties for the considered equation so as to check the possibility that this equation passes the Painlevé test. Secondly, Lie group analysis is studied for finding the symmetries by using Lie classical group analysis method and to obtain its symmetry group, infinitesimal generator, Lie algebra commutation table, and similarity reductions. The vector fields and the symmetry reduction of this equation are calculated with the aid of Lie symmetry analysis. From the similarity reduction equation, some explicit exact solutions are derived. Finally, using the new conservation theorem proposed by Ibragimov [N. H. Ibragimov, A new conservation theorem, J. Math. Anal. Appl. 333 (2007) 311–328], the conservation laws of the aforesaid equation have been constructed.

It has been proposed that f(R) gravity is a viable proxy for General Relativity and a means for an alternative to dark energy. Within this context, we present the analytical solutions to the Bianchi type I universe. For this spacetime, the Killing and homothetic vector fields categorize the space into fourteen classes. Noether’s theorem is used to determine the first integrals associated with each isometry for every class and we use the geometry of the Noether point symmetries of the space to establish potential functions. We also quantize the Lagrangian of the spacetime to derive the Wheeler–DeWitt equation and successively determine invariant solutions for the wave function of the universe from the Lie point symmetries admitted by the equation. Finally, for all cases, exact classical solutions to the field equations are found.

In this paper, we consider linearizability and orbital linearizability properties of the Lotka–Volterra system in the neighborhood of a singular point with eigenvalues 1 and -q. In this paper we give the explicit smooth near-identity change of variables that linearizes or orbital linearizes such Lotka–Volterra system with q ∈ ℕ\{0, 1} being seen that these changes are also valid for q ∈ ℂ\{0, 1}.

We present an equation of the fourth-order which does not possess a second-order Lagrangian and demonstrate by means of the method of reduction of order that one can obtain a first-order Lagrangian for it. This opens the way to quantization through the construction of an Hamiltonian which is suitable to be quantized according to the procedure of Dirac with the correct physical attributes.