Lie-Backlund Symmetries Research Articles

The computation of Lie point symmetry generators, modulational instability, classification of conserved quantities, and explicit power series solutions of the coupled system

The well-known Chaffee–Infante reaction hierarchy is examined in this article along with its reaction–diffusion coupling. It has numerous variety of applications in modern sciences, such as electromagnetic wave fields, fluid dynamics, high-energy physics, ion-acoustic waves in plasma physics, coastal engineering, and optical fibres. The physical processes of mass transfer and particle diffusion might be expressed in this way. The Lie invariance criteria is taken into consideration while we determine the symmetry generators. The suggested approach produces the six dimensional Lie algebra, where translation symmetries in space and time are associated to mass conservation and conservation of energy respectively, the other symmetries are scaling or dilation. Additionally, similarity reductions are performed, and the optimal system of the sub-algebra should be quantified. There are an enormous number of exact solutions can construct for the traveling waves when the governing system is transformed into ordinary differential equations using the similarity transformation technique. The power series approach is also utilized for ordinary differential equations to obtain closed-form analytical solutions for the proposed diffusive coupled system. The stability of the model under the limitations is ensured by the modulation instability analysis. The reaction diffusion hierarchy’s conserved vectors are calculated using multiplier methods using Lie Backlund symmetries. The acquired results are presented graphically in 2-D and 3-D to demonstrate the wave propagation behavior.

Conservation laws of [formula omitted]-dimensional time-fractional diffusion equation

The concept of Lie–Backlund symmetry plays a fundamental role in applied mathematics. It is clear that in order to find conservation laws for a given partial differential equations (PDEs) or fractional differential equations (FDEs) by using Lagrangian function, firstly, we need to obtain the symmetries of the considered equation.Fractional derivation is an efficient tool for interpretation of mathematical methods. Many applications of fractional calculus can be found in various fields of sciences as physics (classic, quantum mechanics and thermodynamics), biology, economics, engineering and etc. So in this paper, we present some effective application of fractional derivatives such as fractional symmetries and fractional conservation laws by fractional calculations. In the sequel, we obtain our results in order to find conservation laws of the time-fractional equation in some special cases.

Invariant subspaces and conditional Lie-Bäcklund symmetries of inhomogeneous nonlinear diffusion equations

The inhomogeneous nonlinear diffusion equation is studied by invariant subspace and conditional Lie-Backlund symmetry methods. It is shown that the equations admit a class of invariant subspaces governed by the nonlinear ordinary differential equations, which is equivalent to a kind of higher-order conditional Lie-Backlund symmetries of the equations. As a consequence, a number of new solutions to the inhomogeneous nonlinear diffusion equations are constructed explicitly or reduced to solving finite-dimensional dynamical systems.

Solutions and symmetry reductions of the n-dimensional non-linear convection-diffusion equations

This paper discusses a wide class of n-dimensional non-linear convection―diffusion equations with source term. It is shown that the radially symmetric equations admit certain types of conditional Lie―Backlund symmetries. As a result, exact solutions and symmetry reductions to 2D dynamical systems of the resulting equations are obtained. Those solutions extend the known ones such as self-similar solutions and instantaneous source-type solutions of the porous medium equation with absorption term. The behaviour of extinction and blow-up to many of the solutions are described.

Nonclassical Approximate Symmetries of Evolution Equations with a Small Parameter

We introduce a method of approximate nonclassical Lie-Backlund symmetries for partial differential equations with a small parameter and discuss applications of this method to finding of approximate solutions both integrable and nonintegrable equations.

A new approach to solve perturbed nonlinear evolution equations through lie-backlund symmetry method

A new method based on Lie-Backlund symmetry method to solve the perturbed nonlinear evolution equations is presented. New approximate solutions of perturbed nonlinear evolution equations stemming from the exact solutions of unperturbed equations are obtained. This method is a generalization of Burde’s Lie point symmetry technique.

Nonclassical potential symmetries and invariant solutions of heat equation

Some nonclassical potential symmetry generators and group-invariant solutions of heat equation and wave equation were determined. It is shown that some new explicit solutions of partial differential equations in conserved form can be constructed by using the nonclassical potential symmetry generators which are derived from their adjoint system. These explicit solutions cannot be obtained by using the Lie or Lie-Backlund symmetry group generators of differential equations.

Painlevé property and conservation laws of multi-component mKdV equations

Abstract It is shown that two classes of n-component mKdV equations are Painleve integrable, and the resonances occur, respectively, at −1,3,4, 0,…,0 n−1 , 1,…,1 n−1 , 5,…,5 n−1 for the geometric n-component mKdV equation which arises from curve motions in Euclidean space, and −1,3,4, 0,…,0 n−1 , 2,…,2 n−1 , 4,…,4 n−1 for another n-component mKdV equation which has an infinite number of Lie Backlund symmetries. It is also shown that the two-component geometric mKdV equation admits an infinite number of conservation laws. Conservation laws for several systems are constructed.

Potential Symmetries and Associated Conservation Laws with Application to Wave Equations

It has been shown that one can generate a class of nontrivial conservation laws for second-order partial differential equations using some recent results dealing with the action of any Lie–Backlund symmetry generator of the equivalentfirst-order system on the respective conservation law. These conservedvectors are nonlocal as they are constructed from associatednonlocal symmetries of the partial differential equation. The method canbe successfully extended to association with ‘genuine’ nonlocal(potential) symmetries. However, it usually involves solving moredifficult systems of partial differential equations which may not alwaysbe easy to uncouple.

Solution of Ordinary Differential Equations with a Large Lie Symmetry Group

The group method of solving ODEs without any quadrature goes back to Lie. In order to apply it, the number of symmetries admitted by a given ODE has to be greater by one than the number of arbitraryconstants in the general solution of the equation. In this paper, we use thetechnique of canonical Lie–Backlund symmetries that makes theproof of the statement concerning integrals of ODEs more evident. The methodis extended to the solution of system of ODEs with a small parameter of arbitrary order.

Relationship between Symmetries andConservation Laws

The fundamental relation between Lie-Backlund symmetry generators andconservation laws of an arbitrary differential equation is derived without regardto a Lagrangian formulation of the differential equation. This relation is used inthe construction of conservation laws for partial differential equations irrespectiveof the knowledge or existence of a Lagrangian. The relation enables one toassociate symmetries to a given conservation law of a differential equation.Applications of these results are illustrated for a range of examples.

Approximate Symmetries and Conservation Laws with Applications

The relationship between the approximateLie-Backlund symmetries and the approximate conservedforms of a perturbed equation is studied. It is shownthat a hierarchy of identities exists by which thecomponents of the approximate conserved vector or theassociated approximate Lie-Backlund symmetries aredetermined by recursive formulas. The results areapplied to certain classes of linear and nonlinear waveequations as well as a perturbed Korteweg-de Vriesequation. We construct approximate conservation laws forthese equations without regard to aLagrangian.

New symmetries and conservation laws for lossless KZK equation

The Khokhlov–Zabolotskaya–Kuznetsov (KZK) equation is widely used to describe intense acoustic beams. No general analytical solution has been obtained for this nonlinear equation, so numerical methods are usually employed. Known analytical approaches are associated with various approximations. An additional powerful mathematical tool is group analysis that enables the finding of general symmetrical properties of differential equations. These symmetries help to generalize known analytical and numerical solutions, to derive new solutions, and to obtain conservation laws. In this work, results of group classification of the KZK equation with arbitrary nonlinear term are presented. It is shown that the KZK equation can be written in Euler–Lagrange form. All the classical Lie symmetries (geometric symmetries) are found and some classes of Lie–Backlund symmetries are considered. The largest number of the symmetries corresponds to the cases of quadratic and cubic nonlinearities. Besides scaling and translation groups of symmetries, there exist additional transformation groups which are not evident from the physical point of view. Examples of obtaining new solutions and deriving reduced equations are presented. New conservation laws for intense acoustic beams are obtained using the Noether theorem. [Work supported by RFBR and CRDF.]

Nonlocal Lie-Backlund Symmetries and Solitary Wave Solutions of the General KdV and mKdV Equations11The project supported by The Zhejiang Training Center of Teacher's College

In this paper, three sets of high-order nonlocal Olver symmetries are obtained from a special nonlocal Lie-Backlund symmetry of the general KdV and general mKdV equation. The soliton solutions of these models can be also obtained from the finite Lie-Backlund transformations.

Conditional Lie-Backlund symmetry and reduction of evolution equations

We suggest a generalization of the notion of invariance of a given partial differential equation with respect to Lie-B\"acklund vector field. Such generalization proves to be effective and enables us to construct principally new Ans\"atze reducing evolution-type equations to several ordinary differential equations. In the framework of the said generalization we obtain principally new reductions of a number of nonlinear heat conductivity equations $u_t=u_{xx}+F(u,u_x)$ with poor Lie symmetry and obtain their exact solutions. It is shown that these solutions can not be constructed by means of the symmetry reduction procedure.

Lie-backlund symmetries and Hamiltonian structure of some new integrable systems

Lie-Backlund symmetries of some new integrable systems, such as (x, t) interchanged Boussinesq equation and a coupled evolutionary system involving three unknowns are obtained via gauge transformations. In each case the Hamiltonian structure is also determined very easily. A new feature of these symmetry transformation is that they depend explicitly on the spectral parameter λ, which in turn induces a new hierarchy of LB symmetries.

Conservation laws and Lie-B�cklund symmetry

A new formula is proposed, expressing the conserved currents of a Lagrangian system of equations in partial derivatives through its Lie-Backlund symmetry. A full description is given of the conserved currents for the Schrodinger, Dirac, Klein-Gordon, and Maxwell equations.

Symmetries for certain coupled nonlinear Schrodinger equations

Generalized symmetries are sought for certain coupled nonlinear Schrodinger equations. It is shown that all Lie-Backlund symmetries but one are equivalent to Lie point symmetries.

Use and construction of potential symmetries

Group-theoretic methods based on local symmetries are useful to construct invariant solutions of PDEs and to linearize nonlinear PDEs by invertible mappings. Local symmetries include point symmetries, contact symmetries and, more generally, Lie-Bäcklund symmetries. An obvious limitation in their utility for particular PDEs is the non-existence of local symmetries. A given system of PDEs with a conserved form can be embedded in a related auxiliary system of PDEs. A local symmetry of the auxiliary system can yield a nonlocal symmetry (potential symmetry) of the given system. The existence of potential symmetries leads to the construction of corresponding invariant solutions as well as to the linearization of nonlinear PDEs by non-invertible mappings. Recent work considers the problem of finding all potential symmetries of given systems of PDEs. Examples include linear wave equations with variable wave speeds as well as nonlinear diffusion, reaction-diffusion, and gas dynamics equations.

Symmetries and integrability of generalized diffusion reaction equations

Using Lie group methods and the Painleve test, the authors analyse nonlinear diffusion reaction equations ut=(D(u)u)+f(u) with power law diffusion coefficients (D approximately uv) and arbitrary nonlinear reaction terms f(u) which have a wide spectrum of applications in many areas of science. The Lie-group-based similarity method leads to a classification of the reaction terms according to its symmetry properties. With the help of the adjoint representation, the optimal system of similarity reductions is calculated. To check the integrability of the partial differential equation, the existence of generalized (Lie-Backlund) symmetries is investigated. Apart from three known cases, no further cases with third-order symmetries exist. Examining the integrability of the second-order ordinary differential equations resulting from the reductions, only a few parameter combinations can be found for which the Painleve property is given. However, they are able to construct unknown integrals of motion for a much larger range of parameter values. From the integrals, exact similarity solutions may be derived. This is demonstrated by examples corresponding to the important moving-wave reduction.