An overview is given of the interplay between symmetries, singularities and integrability and their uses in nonlinear problems arising in Mathematical Physics and Cosmology. A particularly important aspect is the role of nonlocal symmetries in deciding about integrability of complex nonlinear problems which do not apparently admit solutions in closed form. The need for a new approach to the evo...

In this paper, we prove the integrability of the gradient ru to an exponent > 2 near the boundary, u being a weak solution of a nonlinear stationary Navier-Stokes equation under general mixed boundary conditions. As a consequence of this the pressure p belongs to the Campanato space L 2;; (() with := n ?2. Our method of proof relies on an adaption of a technique by Gehring-Giaquinta-Modica (hig...

A master equation expressing the classical integrability of two-dimensional non-linear sigma models is found. The geometrical properties of this equation are outlined. In particular, a closer connection between integrability and T-duality transformations is emphasised. Finally, a whole new class of integrable non-linear sigma models is found and all their corresponding Lax pairs depend on a spe...

in this paper we study the relationships existing between total measurability in variation and gould type fuzzy integrability (introduced and studied in [21]), giving a special interest on their behaviour on atoms and on finite unions of disjoint atoms. we also establish that any continuous real valued function defined on a compact metric space is totally measurable in the variation of a regula...

Abstract Determining whether a dynamical system is integrable generally difficult task which currently done on case by basis requiring large human input. Here we propose and test an automated method to search for the existence of relevant structures, Lax pair connection respectively. By formulating this as optimization problem, are able identify appropriate structures via machine learning techn...

In this paper we explore the general conditions in order that a 2-dimensional natural Hamiltonian system possess a second invariant which is a polynomial in the momenta and is therefore Liouville integrable. We examine the possibility that the invariant is preserved by the Hamiltonian flow on a given energy hypersurface only (weak integrability) and derive the additional requirement necessary t...

The integrability of this system was conjectured in [16] and in the book of V.V. Kozlov [17]. This system appears first in the classification of Birkhoff integrable systems by Kozlov and Treshchev [16]. The classification involves systems with exponential interraction with sufficient number of integrals, polynomial in the momenta. The classification gives necessary conditions for a system with ...

We investigate a generalisation of the three dimensional spring-pendulum system. The problem depends on two real parameters (k, a), where k is the Young modulus of the spring and a describes the nonlinearity of elastic forces. We show that this system is not integrable when k 6= −a. We carefully investigated the case k = −a when the necessary condition for integrability given by the Morales-Ram...

Application of our algebraic approach to Liouville integrable defects is proposed for the sine-Gordon model. Integrability of the model is ensured by the underlying classical r-matrix algebra. The first local integrals of motion are identified together with the corresponding Lax pairs. Continuity conditions imposed on the time components of the entailed Lax pairs give rise to the sewing conditi...