Nonclassical Symmetries Research Articles

Non-classical symmetries and invariant solutions of non-linear Dirac equations

We apply Lie and non-classical symmetry methods to partial differential equations in order to derive solutions of the non-linear Dirac equation corresponding to the Gross–Neveu model in d=(1+1) and d=(2+1) space–time dimensions. For each d, we first identify sub-algebras of the Poincaré–Lie algebra and for each such sub-algebra, we calculate the invariant solution. Non-classical symmetries are also determined and used to derive new solutions for the Gross–Neveu model.

Symmetry solutions for reaction–diffusion equations with spatially dependent diffusivity

Nonclassical and classical symmetry techniques are employed to analyse a reaction–diffusion equation with a cubic source term. Here, the diffusivity (diffusion term) is assumed to be an arbitrary function of the spatial variable. Classification using Lie point and nonclassical symmetries is performed. It turns out that the diffusivity needs to be given as a quadratic function of the spatial variable for the given governing equation to admit nonclassical symmetries. Both nonclassical and classical symmetries are used to construct some group-invariant (exact) solutions. The results are applied to models arising in population dynamics.

Solutions and reductions for radiative energy transport in laser-heated plasma

A full symmetry classification is given for models of energy transport in radiant plasma when the mass density is spatially variable and the diffusivity is nonlinear. A systematic search for conservation laws also leads to some potential symmetries and to an integrable nonlinear model. Classical point symmetries, potential symmetries, and nonclassical symmetries are used to effect variable reductions and exact solutions. The simplest time-dependent solution is shown to be stable and relevant to a closed system.

What symmetries can do for you

Several applications of Lie symmetries and its generalisation are presented: from turning butterflies into tornados, to its applications in epidemics, population dynamics, and ultimately converting classical problems into the quantum realm. Applications of nonclassical symmetries are also illustrated.

Group-Invariant Solutions for the Generalised Fisher Type Equation

In this paper, we construct the group-invariant (exact) solutions for the generalised Fisher type equation using both classical Lie point and the nonclassical symmetry techniques. The generalised Fisher type equation arises in theory of population dynamics. The diffusion term and coefficient of the source term are given as the power law functions of the spatial variable. We introduce the modified Hopf-Cole transformation to simplify a nonlinear second Order Ordinary Equation (ODE) into a solvable linear third order ODE.

Classical and nonclassical symmetries of a nonlinear differential equation for describing waves in a liquid with gas bubbles

In this paper, we consider a nonlinear differential equation for describing nonlinear waves in a liquid with gas bubbles if the liquid viscosity and the interphase heat exchange are accounted for. Classical and nonclassical symmetries of this partial differential equation are investigated. We show that it is invariant under shift transformations in space and time. At an additional restriction on the parameters, this equation is also invariant under the Galilean transformation. Nonclassical symmetries of the equation in question are found by the Bluman-Cole method. Both regular and singular cases of nonclassical symmetries are considered. Five families of nonclassical symmetries admitted by this equation are specified. Invariant reductions corresponding to these families are obtained. With their use, families of exact solutions of the considered equation are found. These solutions are expressed in terms of rational, trigonometric, and special functions.

On symmetry groups of a 2D nonlinear diffusion equation with source

Symmetry analysis of a 2D nonlinear evolutionary equation with mixed spatial derivative and general source term involving the dependent variable and its spatial derivatives is performed. The source terms for which the equation admits nontrivial Lie symmetries are identified for two different forms of the symmetry operator. In one of these cases, the symmetries do not depend on the form of nonlinearities and in the other case, nonlinearities of power, exponential and trigonometric forms are considered. There are no supplementary nonclassical symmetries for the investigated equation. The results reported here generalize the previous results on the 2D heat equation and the 2D Ricci model.

Analytical and numerical studying of the perturbed Korteweg-de Vries equation

The perturbed Korteweg-de Vries equation is considered. This equation is used for the description of one-dimensional viscous gas dynamics, nonlinear waves in a liquid with gas bubbles and nonlinear acoustic waves. The integrability of this equation is investigated using the Painlevé approach. The condition on parameters for the integrability of the perturbed Korteweg-de Vries equation equation is established. New classical and nonclassical symmetries admitted by this equation are found. All corresponding symmetry reductions are obtained. New exact solutions of these reductions are constructed. They are expressed via trigonometric and Airy functions. Stability of the exact solutions of the perturbed Korteweg-de Vries equation is investigated numerically.

Symmetry analysis for a class of nonlinear dispersive equations

A class of dispersive equations is studied within the framework of group analysis of differential equations. The enhanced Lie group classification is achieved. The complete list of equivalence transformations is presented. It is shown that certain equations from the class admit nonclassical reductions. Potential and potential nonclassical symmetries are also considered.

Symmetries, the current function, and exact solutions for Broadwell’s two-dimensional stationary kinetic model

We investigate the Broadwell stationary kinetic model for four velocities on a plane using the current function that satisfies a partial differential equation. For this equation, we evaluate the algebras of classical and nonclassical symmetries and then construct invariant solutions. All classes of solutions describe nonpotential flows. We consider the relation between nonclassical symmetries and previously obtained solutions.

Generalized conditional symmetries and related solutions of the Grad-Shafranov equation

The generalized conditional symmetry (GCS) method is applied to a specific case of the Grad–Shafranov (GS) equation, in cylindrical geometry assuming the existence of an axial symmetry. We investigate the conditions that yield the GS equation admitting a special class of second-order GCSs. The determining system for the unknown arbitrary functions is solved in several special cases and new exact solutions, including solitary waves, different in form and structure from the ones obtained using other nonclassical symmetry methods, are pointed out. Several plots of the level sets or flux surfaces of the new solutions as well as surfaces with vanishing flow are displayed. The obtained solutions can be useful for studying plasma equilibrium, transport phenomena, and magnetohydrodynamic stability.

Unified Mathematical Framework for Slicing and Symmetry Reduction over Event Structures

Nonclassical slicing and symmetry reduction can act as efficient structural abstract methods for pruning state space when dealing with verification problems. In this paper, we mainly address theoretical and algorithmic aspects for nonclassical slicing and symmetry reduction over prime event structures. We propose sliced and symmetric quotient reduction models of event structures and present their corresponding algorithms. To construct the underlying foundation of the proposed methodologies, we introduce strong and weak conflict concepts and a pair of mutually inverse operators and extend permutation group based symmetry notion of event structures. We have established a unified mathematical framework for slicing and symmetry reduction, and further investigated the translation, isomorphism, and equivalence relationship and other related basic facts from a theoretical point of view. The framework may provide useful guidance and theoretical exploration for overcoming verification challenges. This paper also demonstrates their practical applications by two cases.

Symmetries for initial value problems

Abstract In this letter we give a less restrictive condition compared to that given by Zhang and Chen (2010), for first order initial conditions to be recoverable with a particular classical or nonclassical symmetry generator. Examples are provided for the generalised Kuramoto–Sivashinsky equation and a nonlinear diffusion equation with a sink term.

An algorithmic method for showing existence of nontrivial non-classical symmetries of partial differential equations without solving determining equations

In this paper, based on differential characteristic set theory and the associated algorithm (also called Wuʼs method), an algorithmic method is presented to decide on the existence of a nontrivial non-classical symmetry of a given partial differential equation without solving the corresponding nonlinear determining system. The theory and algorithm give a partial answer for the open problem posed by P.A. Clarkson and E.L. Mansfield in [21] on non-classical symmetries of partial differential equations. As applications of our algorithm, non-classical symmetries and corresponding invariant solutions are found for several evolution equations.

Type-II hidden symmetry and nonlinear self-adjointness of Boiti–Leon–Pempinelli equation

This paper focuses on two aspects. Firstly, we convert Boiti–Leon–Pempinelli (BLP) equation to (1+1)-dimensional partial differential equation via similarity transformation, and then analyze hidden symmetry of BLP equations via studying classical and nonclassical symmetries of the (1+1)-dimensional equations. As a byproduct, some new invariant solutions of BLP equations are constructed. Secondly, we show that BLP equation is nonlinearly self-adjoint and give the general formula of conservation laws.

Non-linear time-dependent flow models of third grade fluids: A conditional symmetry approach

In this communication some non-linear flow problems dealing with the unsteady flow of a third grade fluid in porous half-space are analyzed. A new class of closed-form conditionally invariant solutions for these flow models are constructed by using the conditional or non-classical symmetry approach. All possible non-classical symmetries of the model equations are obtained and various new classically invariant solutions have been constructed. The solutions are valid for a half-space and also satisfy the physically relevant initial and the boundary conditions.

Nonclassical Symmetries for a Class of Reaction-Diffusion Equations: the Method of Heir-Equations

The nonclassical symmetries method is applied to a class of reaction-diffusion equations with nonlinear source, i.e. u t =u xx +cu x +R(u, x). Several cases are obtained by using suitable solutions of the heir-equations as described in [M.C. Nucci, Nonclassical symmetries as special solutions of heir-equations, J. Math. Anal. Appl. 279 (2003) 168–179].

Symmetry analysis of a heat conduction model for heat transfer in a longitudinal rectangular fin of a heterogeneous material

We consider a heat conduction model arising in transient heat transfer through longitudinal fins of a heterogeneous (functionally graded) material. In this case, the thermal conductivity depends on the spatial variable. The heat transfer coefficient depends on the temperature and is given by the power law function. The resulting nonlinear partial differential equation is analyzed using both classical and nonclassical symmetry techniques. Both the transient state and the steady state result in a number of exotic symmetries being admitted by the governing equation. Furthermore, nonclassical symmetries are also admitted. Both classical and nonclassical symmetry analysis results in some useful reductions and some remarkable exact solutions are constructed.

Symmetry properties for a generalised thin film equation

Symmetry properties are presented for a fourth-order parabolic equation written in conservation form. It was introduced in the literature as a generalisation of the fourth-order thin film equation. We derive equivalence transformations, Lie symmetries, potential symmetries, non-classical symmetries and potential non-classical symmetries. A chain of such equations is introduced. We conclude by presenting similar results for the third-order equation of this chain.

Nonclassical analysis of the nonlinear Kompaneets equation

The dimensional nonlinear Kompaneets (NLK) equation \({u_{t} = x^{-2}[x^{4}(\alpha{u}_{x} + \beta{u} + \gamma{u}^{2})]_{x}}\) describes the spectra of photons interacting with a rarefied electron gas. Recently, Ibragimov obtained some time-dependent exact solutions for several approximations of this equation. In this paper, we use the nonclassical method to construct time-dependent exact solutions for the NLK equation \({u_{t} = x^{-2}[x^{4}(\alpha{u}_{x} + \gamma{u}^{2})]_{x}}\) for arbitrary constants α > 0, γ > 0. Solutions arising from “nonclassical symmetries” are shown to yield wider classes of time-dependent exact solutions for the NLK equation \({u_{t} = x^{-2}[x^{4}(\alpha{u}_{x} + \gamma{u}^{2})]_{x}}\) beyond those obtained by Ibragimov. In particular, for five classes of initial conditions, each involving two parameters, previously unknown explicit time-dependent solutions are obtained. Interestingly, each of these solutions is expressed in terms of elementary functions. Three of the classes exhibit quiescent behavior, i.e., \({\lim_{t\rightarrow \infty}u(x,t) = 0}\) , and the other two classes exhibit blow-up behavior in finite time. As a consequence, it is shown that the corresponding nontrivial stationary solutions are unstable.