Correspondence theorems for hierarchies of equations of pseudo-spherical type

Hierarchies of evolution equations of pseudo-spherical type are introduced, thereby generalizing the notion of a single equation describing pseudo-spherical surfaces due to S.S. Chern and K. Tenenblat, and providing a connection between differential geometry and the study of hierarchies of equations which are the integrability condition of sl(2,R)-valued linear problems. As an application, it is shown that there exist local correspondences between any two (suitably generic) solutions of arbitrary hierarchies of equations of pseudo-spherical type.     

Non-local symmetries of first-order equations

It is shown how one can transform scalar first-order ordinarydifferential equations which admit non-local symmetries of theexponential type to integrable equations admitting canonicalexponential non-local symmetries. As examples we invoke theAbel equation of the second kind, the Riccati equation and naturalgeneralizations of these. Moreover, our method describes howa double reduction of order for a second-order ordinary differentialequation which admits a two-dimensional Lie algebra of generatorsof point symmetries can be affected if the second-order equationis first reduced in order once by a symmetry which does notspan an ideal of the two-dimensional Lie algebra.     

A new conservation theorem

A general theorem on conservation laws for arbitrary differential equations is proved. The theorem is valid also for any system of differential equations where the number of equations is equal to the number of dependent variables. The new theorem does not require existence of a Lagrangian and is based on a concept of an adjoint equation for non-linear equations suggested recently by the author. It is proved that the adjoint equation inherits all symmetries of the original equation. Accordingly, one can associate a conservation law with any group of Lie, Lie-Bäcklund or non-local symmetries and find conservation laws for differential equations without classical Lagrangians.     

Symmetries of Integrable Difference Equations on the Quad-Graph

This paper describes symmetries of all integrable difference equations that belong to the famous Adler–Bobenko–Suris classification. For each equation, the characteristics of symmetries satisfy a functional equation, which we solve by reducing it to a system of partial differential equations. In this way, all five-point symmetries of integrable equations on the quad-graph are found. These include mastersymmetries, which allow one to construct infinite hierarchies of local symmetries. We also demonstrate a connection between the symmetries of quad-graph equations and those of the corresponding Toda type difference equations.     

Symmetry algebras of linear differential equations

The local symmetries of linear differential equations are investigated by means of proven theorems on the structure of the algebra of local symmetries of translationally and dilatationally invariant differential equations. For a nonparabolic second-order equation, the absence of nontrivial nonlinear local symmetries is proved. This means that the local symmetries reduce to the Lie algebra of linear differential symmetry operators. For the Laplace—Beltrami equation, all local symmetries reduce to the enveloping algebra of the algebra of the conformal group.Tomsk State University. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 92, No. 1, pp. 3–12, July, 1992.     

On the geometry of the Clairin theory of conditional symmetries for higher-order systems of PDEs with applications

This work presents a geometrical formulation of the Clairin theory of conditional symmetries for higher-order systems of partial differential equations (PDEs). We devise methods for obtaining Lie algebras of conditional symmetries from known conditional symmetries, and unnecessary previous assumptions of the theory are removed. As a consequence, new insights into other types of conditional symmetries arise. We then apply the so-called PDE Lie systems to the derivation and analysis of Lie algebras of conditional symmetries. In particular, we develop a method for obtaining solutions of a higher-order system of PDEs via the solutions and geometric properties of a PDE Lie system, whose form gives a Lie algebra of conditional symmetries of the Clairin type. Our methods are illustrated with physically relevant examples such as nonlinear wave equations, the Gauss–Codazzi equations for minimal soliton surfaces, and generalised Liouville equations.     

Weingarten surfaces and nonlinear partial differential equations

The sine-Gordon equation has been known for a long time as the equation satisfied by the angle between the two asymptotic lines on a surface inR ³ with constant Gauss curvature –1. In this paper, we consider the following question: Does any other soliton equation have a similar geometric interpretation? A method for finding all the equations that have such an interpretation using Weingarten surfaces inR ³ is given. It is proved that the sine-Gordon equation is the only partial differential equation describing a class of Weingarten surfaces inR ³ and having a geometricso(3)-scattering system. Moreover, it is shown that the elliptic Liouville equation and the elliptic sinh-Gordon equation are the only partial differential equations describing classes of Weingarten surfaces inR ³ and having geometricso(3,C)-scattering systems.     

Integrable ordinary differential equations on free associative algebras

We consider a classification problem for integrable nonlinear ordinary differential equations with an independent variable belonging to a free associative algebra M. Every equation of this type admits an m×m matrix reduction for an arbitrary m. The existence of symmetries or first integrals belonging to M is used as an integrability criterion. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 122, No. 1, pp. 88–101, January, 1999.     

On the Geometry of Point-Transformation Invariant Class of Third-Order Ordinary Differential Equations

Applying geometric methods, we study a class of third-order ordinary differential equations closed with respect to point transformations. We associate with such an equation the pseudovector fields formed by its coefficients. The equation possesses a maximal algebra of point-transformation symmetries if five pseudovector fields vanish.__________Translated from Matematicheskie Zametki, vol. 77, no. 5, 2005, pp. 719–726.Original Russian Text Copyright ©2005 by V. V. Kartak.     

Euler integral symmetries for the confluent Heun equation and symmetries of the Painlevé equation PV

Euler integral symmetries relate solutions of ordinary linear differential equations and generate integral representations of the solutions in several cases or relations between solutions of constrained equations. These relations lead to the corresponding symmetries of the monodromy matrices for the differential equations. We discuss Euler symmetries in the case of the deformed confluent Heun equation, which is in turn related to the Painlevé equation PV. The existence of symmetries of the linear equations leads to the corresponding symmetries of the Painlevé equation of the Okamoto type. The choice of the system of linear equations that reduces to the deformed confluent Heun equation is the starting point for the constructions. The basic technical problem is to choose the bijective relation between the system parameters and the parameters of the deformed confluent Heun equation. The solution of this problem is quite large, and we use the algebraic computing system Maple for this.     

Pseudospherical Surfaces and Evolution Equations

We consider evolution equations, mainly of type u_t = F(u, u_x,..., ?^ku/?x^k), which describe pseudo-spherical surfaces. We obtain a systematic procedure to determine a linear problem for which a given equation is the integrability condition. Moreover, we investigate how the geometrical properties of surfaces provide analytic information for such equations.     

New origins of hidden symmetries for partial differential equations

Hidden symmetries of differential equations are point symmetries that arise unexpectedly in the increase (equivalently decrease) of order, in the case of ordinary differential equations, and variables, in the case of partial differential equations. The origins of Type II hidden symmetries (obtained via reduction) for ordinary differential equations are understood to be either contact or nonlocal symmetries of the original equation while the origin for Type I hidden symmetries (obtained via increase of order) is understood to be nonlocal symmetries of the original equation. Thus far, it has been shown that the origin of hidden symmetries for partial differential equations is point symmetries of another partial differential equation of the same order as the original equation. Here we show that hidden symmetries can arise from contact and nonlocal/potential symmetries of the original equation, similar to the situation for ordinary differential equations.     

Nonlocal symmetries and recursion operators: Partial differential and differential-difference equations

A systematic method to derive the nonlocal symmetries for partial differential and differential-difference equations with two independent variables is presented and shown that the Korteweg-de Vries (KdV) and Burger's equations, Volterra and relativistic Toda (RT) lattice equations admit a sequence of nonlocal symmetries. An algorithm, exploiting the obtained nonlocal symmetries, is proposed to derive recursion operators involving nonlocal variables and illustrated it for the KdV and Burger's equations, Volterra and RT lattice equations and shown that the former three equations admit factorisable recursion operators while the RT lattice equation possesses (2×2) matrix factorisable recursion operator. The existence of nonlocal symmetries and the corresponding recursion operator of partial differential and differential-difference equations does not always determine their mathematical structures, for example, bi-Hamiltonian representation.     

Nonlocal Symmetries of the Camassa-Holm Type Equations

A class of nonlocal symmetries of the Camassa-Holm type equations with bi-Hamiltonian structures, including the Camassa-Holm equation, the modified Camassa-Holm equation, Novikov equation and Degasperis-Procesi equation, is studied. The nonlocal symmetries are derived by looking for the kernels of the recursion operators and their inverse operators of these equations. To find the kernels of the recursion operators, the authors adapt the known factorization results for the recursion operators of the KdV, modified KdV, Sawada-Kotera and Kaup-Kupershmidt hierarchies, and the explicit Liouville correspondences between the KdV and Camassa-Holm hierarchies, the modified KdV and modified Camassa-Holm hierarchies, the Novikov and Sawada-Kotera hierarchies, as well as the Degasperis-Procesi and Kaup-Kupershmidt hierarchies.     

Pseudo-potentials, nonlocal symmetries and integrability of some shallow water equations

Zero curvature formulations, pseudo-potentials, modified versions, “Miura transformations”, conservation laws, and nonlocal symmetries of the Korteweg–de Vries, Camassa–Holm and Hunter–Saxton equations are investigated from a unified point of view: these three equations belong to a two-parameter family of equations describing pseudo-spherical surfaces, and therefore their basic integrability properties can be studied by geometrical means.      

Isospectral deformation of the reduced quasi-classical self-dual Yang–Mills equation

We derive a new four-dimensional partial differential equation with the isospectral Lax representation by shrinking the symmetry algebra of the reduced quasi-classical self-dual Yang–Mills equation and applying the technique of twisted extensions to the obtained Lie algebra. Then we find a recursion operator for symmetries of the new equation and construct a Bäcklund transformation between this equation and the four-dimensional Martínez Alonso–Shabat equation. Finally, we construct extensions of the integrable hierarchies associated to the hyper-CR equation for Einstein–Weyl structures, the reduced quasi-classical self-dual Yang–Mills equation, the four-dimensional universal hierarchy equation, and the four-dimensional Martínez Alonso–Shabat equation.     

Discrete equation on a square lattice with a nonstandard structure of generalized symmetries

We clarify the integrability nature of a recently found discrete equation on the square lattice with a nonstandard symmetry structure. We find its L-A pair and show that it is also nonstandard. For this discrete equation, we construct the hierarchies of both generalized symmetries and conservation laws. This equation yields two integrable systems of hyperbolic type. The hierarchies of generalized symmetries and conservation laws are also nonstandard compared with known equations in this class.     

On symmetries of stochastic differential equations

This note can be considered as a supplement to article [8]. Its purpose is twofold. First, to show that symmetries of Itô stochastic differential equations form a Lie algebra. Second, to provide more precise formulation of the relation between symmetries of SDEs and symmetries of the associated Fokker–Planck equation. Relation between first integrals of SDEs and symmetries of the associated Fokker–Planck equation is also considered.     

Some Geometric Aspects of Integrability of Differential Equations in Two Independent Variables

The relation between scalar evolution equations which are the integrability condition of sl(2,R)-valued linear problems with parameter (kinematic integrability) and those which possess recursion operators (formal integrability) is studied: using that kinematically integrable equations describe one-parameter families of pseudo-spherical surfaces and vice versa, it is shown that every second order formally integrable evolution equation is kinematically integrable, and that this result cannot be extended as proven to the third-order case.Conservation laws of kinematically integrable equations obtained from their underlying pseudo-spherical structure are compared with the ones one finds from the Riccati equation version of their associated linear problems. Symmetries (generalized/nonlocal) for these equations are also studied, by considering infinitesimal deformations of the associated pseudo-spherical surfaces.Finally, conservation laws for equations describing pseudo-spherical surfaces immersed in a flat three-space are found, and the class of equations describing Calapso–Guichard surfaces is introduced.