Artur Sergyeyev

Why nonlocal recursion operators produce local symmetries: new results and applications

It is well known that integrable hierarchies in (1+1) dimensions are local while the recursion operators that generate these hierarchies usually contain nonlocal terms. We resolve this apparent discrepancy by providing simple and universal sufficient conditions for a (nonlocal) recursion operator in (1+1) dimensions to generate a hierarchy of local symmetries. These conditions are satisfied by virtually all recursion operators known today and are much easier to verify than those found in earlier work. We also give explicit formulae for the nonlocal parts of higher recursion, Hamiltonian and symplectic operators of integrable systems in (1+1) dimensions. Using these two results we prove, under some natural assumptions, the Maltsev–Novikov conjecture stating that higher Hamiltonian, symplectic and recursion operators of integrable systems in (1+1) dimensions are weakly nonlocal, i.e., the coefficients of these operators are local and these operators contain at most one integration operator in each term.

Conservation laws and normal forms of evolution equations

Abstract We study local conservation laws for evolution equations in two independent variables. In particular, we present normal forms for the equations admitting one or two low-order conservation laws. Examples include Harry Dym equation, Korteweg–de Vries-type equations, and Schwarzian KdV equation. It is also shown that for linear evolution equations all their conservation laws are (modulo trivial conserved vectors) at most quadratic in the dependent variable and its derivatives.

Generalized Stäckel transform and reciprocal transformations for finite-dimensional integrable systems

We present a multiparameter generalization of the Stackel transform (the latter is also known as the coupling-constant metamorphosis) and show that under certain conditions this generalized Stackel transform preserves Liouville integrability, noncommutative integrability and superintegrability. The corresponding transformation for the equations of motion proves to be nothing but a reciprocal transformation of a special form, and we investigate the properties of this reciprocal transformation. Finally, we show that the Hamiltonians of the systems possessing separation curves of apparently very different form can be related through a suitably chosen generalized Stackel transform.

Recursion operators for dispersionless integrable systems in any dimension

We present a new approach to construction of recursion operators for multidimensional integrable systems which have a Lax-type representation in terms of a pair of commuting vector fields. It is illustrated by the examples of the Manakov–Santini system which is a hyperbolic system in N dependent and (N + 4) independent variables, where N is an arbitrary natural number, the six-dimensional generalization of the first heavenly equation, the modified heavenly equation and the dispersionless Hirota equation.

A Simple Way of Making a Hamiltonian System Into a Bi-Hamiltonian One

Given a Poisson structure (or, equivalently, a Hamiltonian operator) P, we show that its Lie derivative Lτ(P) along a vector field τ defines another Poisson structure, which is automatically compatible with P, if and only if [Lτ2(P),P]=0, where [⋅,⋅] is the Schouten bracket. This result yields a new local description for the set of all Poisson structures compatible with a given Poisson structure P of locally constant rank such that dim ker P≤1 and leads to a remarkably simple construction of bi-Hamiltonian dynamical systems. A new description for pairs of compatible local Hamiltonian operators of Dubrovin–Novikov type is also presented.

Ordering of two small parameters in the shallow water wave problem

The classical problem of irrotational long waves on the surface of a shallow layer of an ideal fluid moving under the influence of gravity as well as surface tension is considered. A systematic procedure for deriving an equation for surface elevation for a prescribed relation between the orders of the two expansion parameters, the amplitude parameter α and the long wavelength (or shallowness) parameter β, is developed. Unlike the heuristic approaches found in the literature, when modifications are made in the equation for surface elevation itself, the procedure starts from the consistently truncated asymptotic expansions for unidirectional waves, a counterpart of the Boussinesq system of equations for the surface elevation and the bottom velocity, from which the leading-order and higher order equations for the surface elevation can be obtained by iterations. The relations between the orders of the two small parameters are taken in the form β = O(αn) and α = O(βm) with n and m specified to some important particular cases. The analysis shows, in particular, that some evolution equations, proposed before as model equations in other physical contexts (such as the Gardner equation, the modified Korteweg–de Vries (KdV) equation and the so-called fifth-order KdV equation), can emerge as the leading-order equations in the asymptotic expansion for the unidirectional water waves, on equal footing with the KdV equation. The results related to the higher orders of approximation provide a set of consistent higher order model equations for unidirectional water waves which replace the KdV equation with higher order corrections in the case of non-standard ordering when the parameters α and β are not of the same order of magnitude. The shortcomings of certain models used in the literature become apparent as a result of the subsequent analysis. It is also shown that various model equations obtained by assuming a prescribed relation β = O(αn) between the orders of the two small parameters can be equivalently treated as obtained by applying transformations of variables which scale out the parameter β, in favor of α. It allows us to consider the nonlinearity-dispersion balance, epitomized by the soliton equations, as existing for any β, provided that α → 0, but leads to a prescription, in asymptotic terms, of the region of time and space where the equations are valid and so the corresponding dynamics are expected to occur.

Sasa-Satsuma (complex modified Korteweg–de Vries II) and the complex sine-Gordon II equation revisited: Recursion operators, nonlocal symmetries, and more

We present a new symplectic structure and a hereditary recursion operator for the Sasa-Satsuma equation which is widely used in nonlinear optics. Using an integrodifferential substitution relating this equation to a third-order symmetry flow of the complex sine-Gordon II equation enabled us to find a hereditary recursion operator and higher Hamiltonian structures for the latter equation. We also show that both the Sasa-Satsuma equation and the third-order symmetry flow for the complex sine-Gordon II equation are bi-Hamiltonian systems, and we construct several hierarchies of local and nonlocal symmetries for these systems.

Maximal superintegrability of Benenti systems

For a class of Hamiltonian systems, naturally arising in the modern theory of separation of variables, we establish their maximal superintegrability by explicitly constructing the additional integrals of motion.

Recursion operator for the stationary Nizhnik–Veselov–Novikov equation

We present a new general construction of a recursion operator from the zero-curvature representation. Using it, we find a recursion operator for the stationary Nizhnik–Veselov–Novikov equation and present a few low-order symmetries generated with the help of this operator.

A strange recursion operator demystified

We show that a new integrable two-component system of KdV type studied by Karasu (Kalkanli) et al (2004 Acta Appl. Math. 83 85–94) is bi-Hamiltonian, and its recursion operator, which has a highly unusual structure of nonlocal terms, can be written as a ratio of two compatible Hamiltonian operators found by us. Using this we prove that the system in question possesses an infinite hierarchy of local commuting generalized symmetries and conserved quantities in involution, and the evolution systems corresponding to these symmetries are bi-Hamiltonian as well. We also show that upon introduction of suitable nonlocal variables the nonlocal terms of the recursion operator under study can be written in the usual form, with the integration operator D−1x appearing in each term at most once.