Table of contents

The Seventh International Workshop "Group Analysis of Differential Equations and Integrable Systems" (GADEIS-VII) took place at Flamingo Beach Hotel, Larnaca, Cyprus during the period June 15-19, 2014. Fifty nine scientists from nineteen countries participated in the Workshop, and forty one lectures were presented. The Workshop topics ranged from theoretical developments of group analysis of differential equations, hypersymplectic structures, theory of Lie algebras, integrability and superintegrability to their applications in various fields.

The Series of Workshops is a joint initiative by the Department of Mathematics and Statistics, University of Cyprus, and the Department of Applied Research of the Institute of Mathematics, National Academy of Sciences, Ukraine. The Workshops evolved from close collaboration among Cypriot and Ukrainian scientists. The first three meetings were held at the Athalassa campus of the University of Cyprus (October 27, 2005, September 25-28, 2006, and October 4-5, 2007). The fourth (October 26-30, 2008), the fifth (June 6-10, 2010) and the sixth (June 17-21, 2012) meetings were held at the coastal resort of Protaras.

We would like to thank all the authors who have published papers in the Proceedings. All of the papers have been reviewed by at least two independent referees. We express our appreciation of the care taken by the referees. Their constructive suggestions have improved most of the papers. The importance of peer review in the maintenance of high standards of scientific research can never be overstated.

Olena Vaneeva, Christodoulos Sophocleous, Roman Popovych, Vyacheslav Boyko, Pantelis Damianou

All papers published in this volume of Journal of Physics: Conference Series have been peer reviewed through processes administered by the proceedings Editors. Reviews were conducted by expert referees to the professional and scientific standards expected of a proceedings journal published by IOP Publishing.

The algebraic method for computing the complete point symmetry group of a system of differential equations is extended to finding the complete equivalence group of a class of such systems. The extended method uses the knowledge of the corresponding equivalence algebra. Two versions of the method are presented, where the first involves the automorphism group of this algebra and the second is based on a list of its megaideals. We illustrate the megaideal-based version of the method with the computation of the complete equivalence group of a class of nonlinear wave equations with applications in nonlinear elasticity.

Admissible point transformations of classes of rth order linear ordinary differential equations (in particular, the whole class of such equations and its subclasses of equations in the rational form, the Laguerre-Forsyth form, the first and second Arnold forms) are exhaustively described. Using these results, the group classification of such equations is carried out within the algebraic approach in three different ways.

In the singularity analysis of ordinary differential equations it often happens that there are various possible leading-order behaviours. It is observed that the sums of the resonances in certain instances are the same. We illustrate this with certain examples from differential sequences and higher-order equations of Bureau symbol P2. For the Riccati Differential Sequence we prove two generic properties dealing with the occurrence of the coefficients of the leading-order term and the variation of the sum of the resonances. An explanation of observed discrepancies in the cases of fourth- and fifth-order equations of Bureau symbol P2 is presented.

We devise a new method for producing Hamiltonian systems by constructing the corresponding Lax pairs. This is achieved by considering a larger subset of the positive roots than the simple roots of the root system of a simple Lie algebra. We classify all subsets of the positive roots of the root system of type A_n for which the corresponding Hamiltonian systems are transformed, via a simple change of variables, to Lotka-Volterra systems. For some special cases of subsets of the positive roots of the root system of type A_n, we produce new integrable Hamiltonian systems.

We construct integrable discrete nonautonomous quad-equations as Bäcklund autotransformations for known Volterra and Toda type semidiscrete equations, some of which are also nonautonomous. Additional examples of this kind are found by using transformations of discrete equations which are invertible on their solutions. In this way we obtain integrable examples of different types: discrete analogs of the sine-Gordon equation, the Liouville equation and the dressing chain of Shabat. For Liouville type equations we construct general solutions, using a specific linearization. For sine-Gordon type equations we find generalized symmetries, conservation laws and L—A pairs.

Group analysis of the spatially homogeneous and molecular energy dependent Boltzmann equations with source term is carried out. The Fourier transform of the Boltzmann equation with respect to the molecular velocity variable is considered. The correspondent determining equation of the admitted Lie group is reduced to a partial differential equation for the admitted source. The latter equation is analyzed by an algebraic method. A complete group classification of the Fourier transform of the Boltzmann equation with respect to a source function is given. The representation of invariant solutions and corresponding reduced equations for all obtained source functions are also presented.

The classical problem of construction of Gardner's deformations for infinite-dimensional completely integrable systems of evolutionary partial differential equations (PDE) amounts essentially to finding the recurrence relations between the integrals of motion. Using the correspondence between the zero-curvature representations and Gardner deformations for PDE, we construct a Gardner's deformation for the Krasil'shchik-Kersten system. For this, we introduce the new nonlocal variables in such a way that the rules to differentiate them are consistent by virtue of the equations at hand and second, the full system of Krasil'shchik-Kersten's equations and the new rules contains the Korteweg-de Vries equation and classical Gardner's deformation for it.

We describe admissible point transformations in the class of (1+2)-dimensional linear Schrödinger equations with complex potentials. We prove that any point transformation connecting two equations from this class is the composition of a linear superposition transformation of the corresponding initial equation and an equivalence transformation of the class. This shows that the class under study is semi-normalized.

All inequivalent realizations of the Poincaré algebras p(1,1) and p(1, 2) acting in spaces of not more than three variables are constructed. First-order deformations of p(1,1) and p(1,2) are proposed and the generic realizations for the initial and deformed Poincaré algebras are presented.

Laplace-Runge-Lentz (LRL) vector is a cornerstone of celestial mechanics. It also plays an important role in quantum mechanics, being an integral of motion for the Hydrogen atom and some other systems. However, the majority of models of non-relativistic systems admitting LRL vector ignore the spin of orbital particles. In this survey a new collection of QM systems admitting LRL vector with spin is presented. It includes 2d and 3d systems with arbitrary spin, as well as systems of arbitrary dimension with spins 0, 1/2, and 1. All these systems are superintegrable and can be solved exactly. They emulate neutral particles with non-trivial multipole momenta (in particular, the neutron) interacting with a central external field.

We find the equivalence groupoid of a class of (1 + 1)-dimensional second-order evolution equations, which are called generalized potential Burgers equations. This class is related via potentialization with two classes of variable-coefficient generalized Burgers equations. Its equivalence groupoid is of complicated structure and is described via partitioning the entire class into three normalized subclasses such that there are no point transformations between equations from different subclasses. For each of these subclasses we construct its equivalence group of an appropriate kind.

We show that each Saletan (linear) contraction can be realized, up to change of bases of the initial and the target Lie algebras, by a matrix-function that is completely defined by a partition of the dimension of Fitting component of its value at the limit value of the contraction parameter. The codimension of the Fitting component and this partition constitute the signature of the Saletan contraction. We study Saletan contractions with Fitting component of maximal dimension and trivial one-part partition. All contractions of such kind in dimension three are completely classified.

Bäcklund-Darboux transformations are closely related to the integrability and symmetry problems. For the generalized Bäcklund-Darboux transformation (GBDT), we consider conservation laws, rational extensions and bispectrality. We use the case of the nonlinear optics equation (and its auxiliary linear system) as an example.

We study the Lie point symmetries of a general class of partial differential equations (PDE) of second order. An equation from this class naturally defines a second-order symmetric tensor (metric). In the case the PDE is linear on the first derivatives we show that the Lie point symmetries are given by the conformal algebra of the metric modulo a constraint involving the linear part of the PDE. Important elements in this class are the Klein-Gordon equation and the Laplace equation. We apply the general results and determine the Lie point symmetries of these equations in various general classes of Riemannian spaces. Finally we study the type II hidden symmetries of the wave equation in a Riemannian space with a Lorenzian metric.

We consider a class of variable-coefficient mKdV equations. We derive the equivalence transformations in the infinitesimal form and we employ them to construct differential invariants of the respective equivalence algebra. Operators of invariant differentiation are also constructed. Applications, similar to Laplace invariants, are presented.

Group classification of a class of Benjamin-Bona-Mahony (BBM) equations with time dependent coefficients is carried out. Two equivalent lists of equations possessing Lie symmetry extensions are presented: up to point equivalence within the class of BBM equations and without the simplification by equivalence transformations. It is shown that the complete results can be achieved using either the gauging of arbitrary elements of the class by the equivalence transformations or the method of mapping between classes. As by-product of the second approach the complete group classification of a class of variable-coefficient BBM equations with forcing term is derived.

We consider an auxiliary linear problem on the algebra sl(3) with reductions of ₂ × ₂ type introduced recently by Gerdjikov, Mikhailov and Valchev (GMV), and discuss its relation with a Generalized Zakharov-Shabat system having the same type of reduction symmetry. We describe the gauge-equivalent hierarchies and find explicitly the analog of the first nontrivial soliton equation in the hierarchy of equations related to GMV linear problem.

We present an approach to systematic description and classification of solutions of partial differential equations that are obtained by means of reduction of these equations to other equations with smaller number of independent variables. We propose to classify such reductions by means of classification of reduction conditions. The approach is illustrated by an example of the system of d'Alembert and eikonal equations. Solutions of this system were used to outline classification of reductions for the general nonlinear d'Alembert equation, with generalisation to arbitrary Poincaré invariant equations.

In this paper we construct integrable three-dimensional quantum-mechanical systems with magnetic fields, admitting pairs of commuting second-order integrals of motion. The case of Cartesian coordinates is considered. Most of the systems obtained are new and not related to the separation of variables in the corresponding Schrödinger equation.