Lie-Backlund Symmetries Research Articles

New nonlocal symmetries with pseudopotentials

The concept of nonlocal Lie-Backlund symmetries can be generalized by including pseudopotentials. For the KdV, HD and AKNS equations, the authors calculate generalized symmetries of such a kind. The solitary wave solutions are obtained through the transformation of trivial solutions.

Approximate Lie and Lie-Backlund symmetry of the Kuramoto-Shivashinsky equation

Following an idea of Baikov et al. (1989), it has been demonstrated that the non-integrable system the Kuramoto-Shivashinsky equation, does possess approximate Lie point and Lie-Backlund symmetries. In both the cases the authors have obtained the explicit form of the generators as a series in the small parameter in . Lie point symmetry considerations lead to a similarity form which finally leads to a perturbed ordinary differential equation.

The Lie-Backlund algebra for the general underdetermined equation ur=f(x,u,.,ur-1, nu ,., nu k)

The Lie-Backlund symmetry algebra for the equation ur=f(x,u, ., ur-1, nu ,., nu k) is established.

Geometry of Virasoro constraints in nonperturbative 2-d quantum gravity

The string equation and the Virasoro constraints for arbitrary hermitian multimatrix models are derived using the Lie-Backlund symmetries of the generalised KdV equations. From this point of view the origin of the string equation and the meaning of the Virasoro constraints are explained. Some speculation about the appearance of extra constraints, which we suspect to be theW-constraints, is also given.

On the Non‐Local Lie Backlund Symmetries and Prolongation Structure for a New Integrable System

On the Non‐Local Lie Backlund Symmetries and Prolongation Structure for a New Integrable System

On the Hamiltonian structure and generalised Lie-Backlund symmetries of Langmuir solitons

The structure of generalised Lie-Backlund symmetries for the coupled equations of Langmuir solitons are analysed in detail. The form of these symmetries, when compared with those of the conservation laws, yields the first symplectic form. The spectra gradient method is then seen to generate the recursion operator for these symmetries which, on factorisation, leads immediately to the second Hamiltonian structure. The same recursion operator, when used along with the (x,t)-dependent symmetries, yields a new class of generalised symmetries for the equations under consideration. Lastly it is observed that these symmetries are in involution with respect to a Jacobi bracket.

Symmetry properties and bi-Hamiltonian structure of the Toda lattice

We carry out a systematic analysis of the Toda lattice equations developing a method which extends the symmetry approach formalism to discrete one-dimensional systems. We find a hereditary operator which admits a symplectic-implectic factorization. As a consequence of this property, we derive the Hamiltonian and the bi-Hamiltonian structure, together with the constants of motion and a set of infinitely-many commuting Lie-Backlund symmetries of the Toda chain.

Painlev� analysis, conservation laws, and symmetry of perturbed nonlinear equations

We consider the Lie-Backlund symmetries and conservation laws of a perturbed KdV equation and NLS equation. The arbitrary coefficients of the perturbing terms can be related to the condition of existence of nontrivial LB symmetry generator. When the perturbed KdV equation is subjected to Painleve analysisa la Weiss, it is found that the resonance position changes compared to the unperturbed one. We prove the compatibility of the overdetermined set of equations obtained at the different stages of recursion relations, at least for one branch. All other branches are also indicated and difficulties associated them are discussed considering the perturbation parametere to be small. We determine the Lax pair for the aforesaid branch through the use of Schwarzian derivative. For the perturbed NLS equation we determine the conservation laws following the approach of Chen and Liu. From the recurrence of these conservation laws a Lax pair is constructed. But the Painleve analysis does not produce a positive answer for the perturbed NLS equation. So here we have two contrasting examples of perturbed nonlinear equations: one passes the Painleve test and its Lax pair can be found from the analysis itself, but the other equation does not meet the criterion of the Painleve test, though its Lax pair is found in another way.

Integrable flow equations that incorporate spatial heterogeneity

Exactly solvable models are constructed in the Darcy-Buckingham approach to unsaturated flow in porous media with continuous spatial variability. The steady soil water potential distribution, for evaporation from a scale-heterogeneous soil with water table, is given explicitly. The cumulative infiltration predicted by a scale-heterogeneous Green-Ampt model is shown to be inconsistent with that expected from a temporal power series solution of the general flow equation. New integrable forms of the unsaturated heterogeneous flow equation are built up from those exactly solvable forms of the homogeneous flow equation which possess special Lie-Backlund symmetry groups.

Non-integrability of non-linear diffusion-convection equations in two-spatial dimensions

It is generally accepted that integrable partial differential time evolution equations possess Lie-Backlund symmetries of arbitrarily high finite order. It has already been established that in (1+1) dimensions the full class of nonlinear Fokker-Planck (convection-diffusion) equations having this property consists of the known integrable examples. These are closely related either to the Burgers equations or to the Fokas-Yortsos-Rosen equation, both of which have been applied to unsaturated flow in porous media. Here the author shows that higher-order Lie-Backlund symmetries do not exist for any examples of the nonlinear Fokker-Planck equation in (1+2) (one time and two space) dimensions.

Bi-Hamiltonian structure and Lie-Backlund symmetries for a modified Harry-Dym system

The authors have obtained the complete Lie-Backlund symmetry for a modified Harry-Dym system and hence deduced the bi-Hamiltonian structure associated with it. It is shown that these Lie-Backlund symmetries generate the recursion operator inherent in the theory and the conserved quantities can also be computed according to the Dorfman prescription.

Bäcklund Transformations and Recursion Operators via Symmetry

In this paper we study interrelations between the Lax-formulation, Backlund transformations and recursion operators. It is shown that different integrability conditions can be connected via Lie-Backlund symmetries of the Lax-formulation. This is then used to derive Backlund transformations and recursion operators for various equations.

Laxpair, Lie-Backlund Symmetry and Hereditary Operator for the Thompson Equation

We have obtained the complete Lie-Backlund symmetry and the hereditary operator for the Thompson equation in the Lie-Ovvsianikov framework. Lastly we write down the Lax Pair for the equation.

Symmetry properties and bi-Hamiltonian structure of the Toda lattice

We carry out a systematic analysis of the Toda lattice equations developing a method which extends the symmetry approach formalism to discrete one-dimensional systems. We find a hereditary operator which admits a symplectic-implectic factorization. As a consequence of this property, we derive the Hamiltonian and the bi-Hamiltonian structure, together with the constants of motion and a set of infinitely-many commuting Lie-Backlund symmetries of the Toda chain.

On the Lie-Backlund symmetry of linear ordinary differential equations

The symmetry of a second-order linear ordinary differential equation and a system of coupled linear ordinary differential equations is investigated. The equations for characteristic functions, which determine the coordinates of the infinitesimal generators of the Lie-Backlund transformations, are found. General solutions of equations for characteristic functions are constructed and the form of infinitesimal generators is obtained.

Elliptic Solutions, Recursion Operators and Complete Lie-Backlund Symmetry for the Harry-Dym Equation

We have made an exhaustive analysis for the equation σt = σ3σκκκ - ασ3σκ, which in the special case reduces to the Harry-Dym equation. First of all we have deduced the Lie point symmetries and the corresponding ordinary differential equation, through the similarity forms. Next the extended Lie-Backlund type generators are deduced. Our equation possesses the special feature of having explicit σ dependent function in the coefficient of highest derivative term, in contrast to the cases discussed by Fokas et al. From the structure of ηI, ηII, ηIII it is easy to deduce the recursion operator Δ. In the second part we deal with the cnoidal wave like solutions. From the Fourier spectrum analysis we show that a cnoidal wave breaks into several ordinary solitary waves.

Complete integrability of the Kortweg-de Vries equation under perturbation around its solution: Lie-Backlund symmetry approach

It is shown that when the Korteweg-de Vries equation is perturbed about a particular solution the resulting evolution equations to all order of perturbations admit infinitely many Lie-Backlund symmetries. The corresponding commuting constants of motion are derived and thereby complete integrability can be established.

Lie-Backlund groups and the linearisation of differential equations

It is shown that groups of Lie-Backlund (LB) transformations which depend on non-local variables are related by a change of variables to the LB tangent transformations of Ibragimov and Anderson (1979), involving no more than arbitrary-order derivatives. The transformation of any LB symmetry operator by an invertible change of variables is discussed. It is pointed out that once a differential equation admits an LB operator, then a large number of 'secondary' equations will admit the same operator. The LB theory involving non-local variables and the notion of secondary equations are used to characterise group theoretically the linearisation of the Burgers equation, ut+uux-uxx=0, and of the ODE uxx+ omega 2(x)u+Ku-3=0.