Journal of Nonlinear Mathematical Physics

Volume 18, Issue 1, March 2011
Letter to Editor

1. The Periodic μ-b-Equation and Euler Equations on the Circle

Martin Kohlmann
Pages: 1 - 8
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)...
Research Article

2. Non-Isospectral 1+1 Hierarchies Arising from a Camassa Holm Hierarchy in 2+1 Dimensions

P. G. Estévez, J. D. Lejarreta, C. Sardón
Pages: 9 - 28
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...
Research Article

3. Integrability of Lie Systems through Riccati Equations

José F. Cariñena, Javier de Lucas
Pages: 29 - 54
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...
Research Article

4. Uq(𝒢^) Heisenberg Families

A. Zuevsky
Pages: 55 - 64
We explicitly describe Heisenberg families of elements in an arbitrary grading subspaces of the quantized universal enveloping algebra Uq(𝒢^) of an affine Kac–Moody algebra 𝒢^ in the Drinfeld formulation.
Research Article

5. Massless Thirring Model in Canonical Quantization Scheme

S. E. Korenblit, V. V. Semenov
Pages: 65 - 74
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...
Research Article

6. Folding Transformations and HKY Mappings

B. Grammaticos, A. Ramani, R. Willox
Pages: 75 - 85
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...
Research Article

7. A Route to Routh — The Classical Setting

Ladislav Adamec
Pages: 87 - 107
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,...
Research Article

8. Geometric Realization of the Two-Point Velocity Correlation Tensor for Isotropic Turbulence

Vladimir N. Grebenev, Martin Oberlack
Pages: 109 - 120
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...
Research Article

9. The Inhomogeneous Invariance Quantum Group of Q-Deformed Boson Algebra with Continuous Parameters

Azmi Ali Altintas, Metin Arik, Ali Serdar Arikan
Pages: 121 - 125
We present a q-deformed boson algebra using continuous momentum parameters and investigate its inhomogeneous invariance quantum group.
Research Article

10. On Edge Waves in Stratified Water Along a Sloping Beach

Raphael Stuhlmeier
Pages: 127 - 137
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...
Research Article

11. Global Existence and Blow-Up Phenomena for the Periodic Hunter–Saxton Equation with Weak Dissipation

Xuemei Wei, Zhaoyang Yin
Pages: 139 - 149
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...
Research Article

12. Derivations of the 3-Lie Algebra Realized by gl(n, ℂ)

Ruipu Bai, Jinxiu Wang, Zhenheng Li
Pages: 151 - 160
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...
Research Article

13. Recursion Operators for KP, mKP and Harry Dym Hierarchies

Jipeng Cheng, Lihong Wang, Jingsong He
Pages: 161 - 178
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...