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Register a new userOn the complete integrability of a nonlinear oscillator from group theoretical perspective
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In this paper, we investigate the integrability aspects of a physically important nonlinear oscillator which lacks sufficient number of Lie point symmetries but can be integrated by quadrature. We explore the hidden symmetry, construct a second integral and derive the general solution of this oscillator by employing the recently introduced $lambda$-symmetry approach and thereby establish the complete integrability of this nonlinear oscillator equation from a group theoretical perspective.
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We show how the change from Eulerian to Lagrangian coordinates for the two-component Camassa-Holm system can be understood in terms of certain reparametrizations of the underlying isospectral problem. The respective coordinates correspond to different normalizations of an associated first order system. In particular, we will see that the two-component Camassa-Holm system in Lagrangian variables is completely integrable as well.
Oblique propagation of magnetohydrodynamic waves in warm plasmas is described by a modified vector derivative nonlinear Schroedinger equation, if charge separation in Poissons equation and the displacement current in Amperes law are properly taken into account. This modified equation cannot be reduced to the standard derivative nonlinear Schroedinger equation and hence its possible integrability and related properties need to be established afresh. Indeed, the new equation is shown to be integrable by the existence of a bi--Hamiltonian structure, which yields the recursion operator needed to generate an infinite sequence of conserved densities. Some of these have been found explicitly by symbolic computations based on the symmetry properties of the new equation. Since the new equation includes as a special case the derivative nonlinear Schroedinger equation, the recursion operator for the latter one is now readily available.
A space discretization of an integrable long wave-short wave interaction model, called the Yajima-Oikawa system, was proposed in the recent paper arXiv:1509.06996 using the Hirota bilinear method (see also https://link.aps.org/doi/10.1103/PhysRevE.91.062902). In this paper, we propose a Lax-pair representation for the discrete Yajima-Oikawa system as well as its multicomponent generalization also considered in arXiv:1509.06996 and prove that it has an infinite number of conservation laws. We also derive the next higher flow of the discrete Yajima-Oikawa hierarchy, which generalizes a modified version of the Volterra lattice. Relations to two integrable discrete nonlinear Schrodinger hierarchies, the Ablowitz-Ladik hierarchy and the Konopelchenko-Chudnovsky hierarchy, are clarified.
Silicene, as the silicon analog of graphene, has been successfully fabricated by epitaxial growing on various substrates. Similar to free-standing graphene, free-standing silicene possesses a honeycomb structure and Dirac-cone-shaped energy band, resulting in many fascinating properties such as high carrier mobility, quantum spin Hall effect, quantum anomalous Hall effect, and quantum valley Hall effect. The maintenance of the honeycomb crystal structure and the Dirac cone of silicene is crucial for observation of its intrinsic properties. In this review, we systematically discuss the substrate effects on the atomic structure and electronic properties of silicene from a theoretical point of view, especially focusing on the changes of the Dirac cone.
R. Hirota and K. Kimura discovered integrable discretizations of the Euler and the Lagrange tops, given by birational maps. Their method is a specialization to the integrable context of a general discretization scheme introduced by W. Kahan and applicable to any vector field with a quadratic dependence on phase variables. According to a proposal by T. Ratiu, discretizations of the Hirota-Kimura type can be considered for numerous integrable systems of classical mechanics. Due to a remarkable and not well understood mechanism, such discretizations seem to inherit the integrability for all algebraically completely integrable systems. We introduce an experimental method for a rigorous study of integrability of such discretizations. Application of this method to the Hirota-Kimura type discretization of the Clebsch system leads to the discovery of four functionally independent integrals of motion of this discrete time system, which turn out to be much more complicated than the integrals of the continuous time system. Further, we prove that every orbit of the discrete time Clebsch system lies in an intersection of four quadrics in the six-dimensional phase space. Analogous results hold for the Hirota-Kimura type discretizations for all commuting flows of the Clebsch system, as well as for the $so(4)$ Euler top.
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