Journal of Nonlinear Mathematical Physics

In this paper, we study the μ-variant of the periodic b-equation and show that this equation can be realized as a metric Euler equation on the Lie group Diff^∞(𝕊) if and only if b = 2 (for which it becomes the μ-Camassa–Holm equation). In this case, the inertia operator generating the metric on Diff^∞(𝕊) is given by . In contrast, the μ-Degasperis–Procesi equation (obtained for b = 3) is not a metric Euler equation on Diff^∞(𝕊) for any regular inertia operator . The paper generalizes some recent results of [13, 16, 24].

The non-isospectral problem (Lax pair) associated with a hierarchy in 2 + 1 dimensions that generalizes the well known Camassa–Holm hierarchy is presented. Here, we have investigated the non-classical Lie symmetries of this Lax pair when the spectral parameter is considered as a field. These symmetries can be written in terms of five arbitrary constants and three arbitrary functions. Different similarity reductions associated with these symmetries have been derived. Of particular interest are the reduced hierarchies whose 1 + 1 Lax pair is also non-isospectral.

Integrability conditions for Lie systems are related to reduction or transformation processes. We here analyze a geometric method to construct integrability conditions for Riccati equations following these approaches. Our procedure provides us with a unified geometrical viewpoint that allows us to analyze some previous works on the topic and explain new properties. Moreover, this new approach can be straightforwardly generalized to describe integrability conditions for any Lie system. Finally we show the usefulness of our treatment in order to study the problem of the linearizability of Riccati equations.

We explicitly describe Heisenberg families of elements in an arbitrary grading subspaces of the quantized universal enveloping algebra of an affine Kac–Moody algebra in the Drinfeld formulation.

It is shown that the exact solubility of the massless Thirring model in the canonical quantization scheme originates from the intrinsic hidden linearizability of its Heisenberg equations in the method of dynamical mappings. The corresponding role of inequivalent representations of free massless Dirac field and appearance of Schwinger terms are elucidated.

We present a new method for the derivation of mappings of HKY type. These are second-order mappings which do not have a biquadratic invariant like the QRT mappings, but rather an invariant of degree higher than two in at least one of the variables. Our method is based on folding transformations which exist for some discrete Painlevé equations. They are transformations which relate the variable of a discrete Painlevé equation to the square of the variable of some other one. By considering the autonomous limit of these relations we derive folding-like transformations which relate QRT mappings to HKY ones. We construct the invariants of the latter mappings and show how they can be extended beyond the ones given by the strict application of the folding transformation.

There is a well known principle in classical mechanic stating that a variational problem independent of a configuration space variable w (so called cyclic variable), but dependent on its velocity w' can be expressed without both w and w'. This principle is known as the Routh reduction.

In this paper, we start to develop a purely geometric approach to this reduction. We do not limit ourselves to rather special problems of mechanics and in a certain sense we are able to obtain explicit formulae for the reduced variational integral.

A new geometric view of homogeneous isotropic turbulence is contemplated employing the two-point velocity correlation tensor of the velocity fluctuations. We show that this correlation tensor generates a family of pseudo-Riemannian metrics. This enables us to specify the geometry of a singled out Eulerian fluid volume in a statistical sense. We expose the relationship of some geometric constructions with statistical quantities arising in turbulence.

We present a q-deformed boson algebra using continuous momentum parameters and investigate its inhomogeneous invariance quantum group.

Building on previous investigations, we show that Gerstner's famous deep water wave and the related edge wave propagating along a sloping beach, found within the context of water of constant density, can both be adapted to provide explicit free surface flows in incompressible fluids with arbitrary density stratification.

In this paper, we study the periodic Hunter–Saxton equation with weak dissipation. We first establish the local existence of strong solutions, blow-up scenario and blow-up criteria of the equation. Then, we investigate the blow-up rate for the blowing-up solutions to the equation. Finally, we prove that the equation has global solutions.

This paper studies structures of the 3-Lie algebra M realized by the general linear Lie algebra gl(n, ℂ). We show that M has only one nonzero proper ideal. We then give explicit expressions of both derivations and inner derivations of M. Finally, we investigate substructures of the (inner) derivation algebra of M.

In this paper, we give a unified construction of the recursion operators from the Lax representation for three integrable hierarchies: Kadomtsev–Petviashvili (KP), modified Kadomtsev–Petviashvili (mKP) and Harry Dym under n-reduction. This shows a new inherent relationship between them. To illustrate our construction, the recursion operator are calculated explicitly for 2-reduction and 3-reduction.