Vladimir S. Novikov

Generalizations of the Camassa?Holm equation

We classify generalized Camassa–Holm-type equations which possess infinite hierarchies of higher symmetries. We show that the obtained equations can be treated as negative flows of integrable quasi-linear scalar evolution equations of orders 2, 3 and 5. We present the corresponding Lax representations or linearization transformations for these equations. Some of the obtained equations seem to be new.

Perturbative symmetry approach

The aim of our paper is to formulate a perturbative version of the symmetry approach in the symbolic representation and to generalize it in order to make it suitable for the study of nonlocal and non-evolution equations. Our formalism is the development and incorporation of the perturbative approach of Zakharov and Schulman, the symbolic method of Sanders and Wang and the standard symmetry approach of Shabat et al. We apply our theory to describe integrable generalizations of the Benjamin-Ono type equations and to isolate integrable cases of the Camassa-Holm type equations.

On Classification of Integrable Nonevolutionary Equations

We study partial differential equations of second order (in time) that possess a hierarchy of infinitely many higher symmetries. The famous Boussinesq equation is a member of this class after the extension of the differential polynomial ring. We develop the perturbative symmetry approach in symbolic representation. Applying it, we classify the homogeneous integrable equations of fourth and sixth order (in the space derivative) equations, as well as we have found three new tenth-order integrable equations. To prove the integrability we provide the corresponding bi-Hamiltonian structures and recursion operators.

Integrable equations in 2 + 1 dimensions: deformations of dispersionless limits

We classify integrable third-order equations in 2 + 1 dimensions which generalize the examples of Kadomtsev–Petviashvili, Veselov–Novikov and Harry Dym equations. Our approach is based on the observation that dispersionless limits of integrable systems in 2 + 1 dimensions possess infinitely many multi-phase solutions coming from the so-called hydrodynamic reductions. In this paper, we adopt a novel perturbative approach to the classification problem. Based on the method of hydrodynamic reductions, we first classify integrable quasilinear systems which may (potentially) occur as dispersionless limits of soliton equations in 2 + 1 dimensions. To reconstruct dispersive deformations, we require that all hydrodynamic reductions of the dispersionless limit be inherited by the corresponding dispersive counterpart. This procedure leads to a complete list of integrable third-order equations, some of which are apparently new.

On a functional equation related to the intermediate long wave equation

We resolve an open problem stated by Ablowitz et al (1982 J. Phys. A: Math. Gen. 15 781) concerning the integral operator appearing in the intermediate long wave equation. We explain how this is resolved using the perturbative symmetry approach introduced by one of us with Mikhailov. By solving a certain functional equation, we prove that the intermediate long wave equation and the Benjamin–Ono equation are the unique integrable cases within a particular class of integro-differential equations. Furthermore, we explain how the perturbative symmetry approach is naturally extended to treat equations on a periodic domain.

Generalizations of the short pulse equation

We classify integrable scalar polynomial partial differential equations of second order generalizing the short pulse equation.

Towards the classification of integrable differential–difference equations in 2 + 1 dimensions

We address the problem of classification of integrable differential–difference equations in 2 + 1 dimensions with one/two discrete variables. Our approach is based on the method of hydrodynamic reductions and its generalization to dispersive equations as proposed in Ferapontov et al (2009 J. Phys. A: Math. Theor. 42 035211, 2009 J. Phys. A: Math. Theor. 42 345205). We obtain a number of classification results of scalar integrable equations including that of the intermediate long wave and Toda type.

Symmetry Structure of Integrable Nonevolutionary Equations

We study a class of evolutionary partial differential systems with two components related to second order (in time) nonevolutionary equations of odd order in spatial variable. We develop the formal diagonalization method in symbolic representation, which enables us to derive an explicit set of necessary conditions of existence of higher symmetries. Using these conditions we globally classify all such homogeneous integrable systems, i.e., systems which possess a hierarchy of infinitely many higher symmetries.

On classification of integrable Davey–Stewartson type equations

This paper is devoted to the classification of integrable Davey-Stewartson type equations. A list of potentially deformable dispersionless systems is obtained through the requirement that such systems must be generated by a polynomial dispersionless Lax pair. A perturbative approach based on the method of hydrodynamic reductions is employed to recover the integrable systems along with their Lax pairs. Some of the found systems seem to be new.

On the classification of scalar evolutionary integrable equations in 2 + 1 dimensions

We consider evolutionary equations of the form ut = F(u, w) where w=Dx−1Dyu is the nonlocality, and the right hand side F is polynomial in the derivatives of u and w. The recent paper [Ferapontov, Moro, and Novikov, J. Phys. A: Math. Theor. 52, 18 (2009)] provides a complete list of integrable third order equations of this kind. Here we extend the classification to fifth order equations. Besides the known examples of Kadomtsev–Petviashvili, Veselov–Novikov, and Harry Dym equations, as well as fifth order analogs and modifications thereof, our list contains a number of equations which are apparently new. We conjecture that our examples exhaust the list of scalar polynomial integrable equations with the nonlocality w. The classification procedure consists of two steps. First, we classify quasilinear systems which may (potentially) occur as dispersionless limits of integrable scalar evolutionary equations. After that we reconstruct dispersive terms based on the requirement of the inheritance of hydrodynamic ...