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Vol 74, No 2 (2019)

Year: 2019
Articles: 7
URL: https://journal-vniispk.ru/0042-1316/issue/view/7509

Polynomial non-integrability of magnetic billiards on the sphere and the hyperbolic plane

Bialy M.L., Mironov A.E.

Abstract

Magnetic billiards in a convex domain with smooth boundary on a constant-curvature surface in a constant magnetic field is considered in this paper. The question of the existence of an integral of motion which is a polynomial in the components of the velocity is investigated. It is shown that if such an integral exists, then the boundary of the domain defines a non-singular algebraic curve in $\mathbb{C}^3$. It is also shown that for a domain other than a geodesic disk, magnetic billiards does not admit a polynomial integral for all but perhaps finitely many values of the magnitude of the magnetic field. To prove our main theorems a new dynamical system, ‘outer magnetic billiards’, on a constant-curvature surface is introduced, a system ‘dual’ to magnetic billiards. By passing to this dynamical system one can apply methods of algebraic geometry to magnetic billiards.Bibliography: 30 titles.

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Uspekhi Matematicheskikh Nauk. 2019;74(2):3-26

3-26

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The finite-gap method and the periodic NLS Cauchy problem of anomalous waves for a finite number of unstable modes

Grinevich P.G., Santini P.M.

Abstract

The focusing non-linear Schrödinger (NLS) equation is the simplest universal model describing the modulation instability (MI) of quasimonochromatic waves in weakly non-linear media, and MI is considered to be the main physical mechanism for the appearance of anomalous (rogue) waves (AWs) in nature. In this paper the finite-gap method is used to study the NLS Cauchy problem for generic periodic initial perturbations of the unstable background solution of the NLS equation (here called the Cauchy problem of AWs) in the case of a finite number $N$ of unstable modes. It is shown how the finite-gap method adapts to this specific Cauchy problem through three basic simplifications enabling one to construct the solution, to leading and relevant order, in terms of elementary functions of the initial data. More precisely, it is shown that, to leading order, i) the initial data generate a partition of the time axis into a sequence of finite intervals, ii) in each interval $I$ of the partition only a subset of ${\mathscr N}(I)\leqslant N$ unstable modes are ‘visible’, and iii) for $t\in I$ the NLS solution is approximated by the ${\mathscr N}(I)$-soliton solution of Akhmediev type describing for these ‘visible’ unstable modes a non-linear interaction with parameters also expressed in terms of the initial data through elementary functions. This result explains the relevance of the $m$-soliton solutions of Akhmediev type with $m\leqslant N$ in the generic periodic Cauchy problem of AWs in the case of a finite number $N$ of unstable modes.Bibliography: 118 titles.

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Uspekhi Matematicheskikh Nauk. 2019;74(2):27-80

27-80

(Rus)

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Real-normalized differentials: limits on stable curves

Grushevsky S., Krichever I.M., Norton C.

Abstract

We study the behaviour of real-normalized (RN) meromorphic differentials on Riemann surfaces under degeneration. We describe all possible limits of RN differentials on any stable curve. In particular we prove that the residues at the nodes are solutions of a suitable Kirchhoff problem on the dual graph of the curve. We further show that the limits of zeros of RN differentials are the divisor of zeros of a twisted differential — an explicitly constructed collection of RN differentials on the irreducible components of the stable curve, with higher order poles at some nodes. Our main tool is a new method for constructing differentials (in this paper, RN differentials, but the method is more general) on smooth Riemann surfaces, in a plumbing neighbourhood of a given stable curve. To accomplish this, we think of a smooth Riemann surface as the complement of a neighbourhood of the nodes in a stable curve, with boundary circles identified pairwise. Constructing a differential on a smooth surface with prescribed singularities is then reduced to a construction of a suitable normalized holomorphic differential with prescribed ‘jumps’ (mismatches) along the identified circles (seams). We solve this additive analogue of the multiplicative Riemann–Hilbert problem in a new way, by using iteratively the Cauchy integration kernels on the irreducible components of the stable curve, instead of using the Cauchy kernel on the plumbed smooth surface. As the stable curve is fixed, this provides explicit estimates for the differential constructed, and allows a precise degeneration analysis.Bibliography: 22 titles.

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Uspekhi Matematicheskikh Nauk. 2019;74(2):81-148

81-148

(Rus)

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Trace formula for the magnetic Laplacian

Kordyukov Y.A., Taimanov I.A.

Abstract

The Guillemin–Uribe trace formula is a semiclassical version of the Selberg trace formula and the more general Duistermaat–Guillemin formula for elliptic operators on compact manifolds, which reflects the dynamics of magnetic geodesic flows in terms of eigenvalues of a natural differential operator (the magnetic Laplacian) associated with the magnetic field. This paper gives a survey of basic notions and results related to the Guillemin–Uribe trace formula and provides concrete examples of its computation for two-dimensional constant curvature surfaces with constant magnetic fields and for the Katok example.Bibliography: 53 titles.

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Uspekhi Matematicheskikh Nauk. 2019;74(2):149-186

149-186