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Symplectic mappings are discrete-time analogs of Hamiltonian systems. They appear in many areas of physics, including, for example, accelerators, plasma, and fluids. Integrable mappings, a subclass of symplectic mappings, are equivalent to a Twist map, with a rotation number, constant along the phase trajectory. In this letter, we propose a succinct expression to determine the rotation number and present two examples. Similar to the period of the bounded motion in Hamiltonian systems, the rotation number is the most fundamental property of integrable maps and it provides a way to analyze the phase-space dynamics.
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The Q1 lattice equation, a member in the Adler-Bobenko-Suris list of 3D consistent lattices, is investigated. By using the multidimensional consistency, a novel Lax pair for Q1 equation is given, which can be nonlinearised to produce integrable symplectic maps. Consequently, a Riemann theta function expression for the discrete potential is derived with the help of the Baker-Akhiezer functions. This expression leads to the algebro-geometric integration of the Q1 lattice equation, based on the commutativity of discrete phase flows generated from the iteration of integrable symplectic maps.
Based on the notion of Darboux-KP chain hierarchy and its invariant submanifolds we construct some class of constraints compatible with integrable lattices. Some simple examples are given.
In this short note we reconsider the integrable case of the Hamiltonian N-species Volterra system, as it has been introduced by Vito Volterra in 1937. In the first part, we discuss the corresponding conserved quantities, and comment about the solutions of the equations of motion. In the second part we focus our attention on the properties of the simplest model, in particular on period and frequencies of the periodic orbits. The discussion and the results presented here are to be viewed as a complement to a more general work, devoted to the construction of a global stationary state model for a sustainable economy in the Hamiltonian formalism.
We study relations between the eigenvectors of rational matrix functions on the Riemann sphere. Our main result is that for a subclass of functions that are products of two elementary blocks it is possible to represent these relations in a combinatorial-geometric way using a diagram of a cube. In this representation, vertices of the cube represent eigenvectors, edges are labeled by differences of locations of zeroes and poles of the determinant of our matrix function, and each face corresponds to a particular choice of a coordinate system on the space of such functions. Moreover, for each face this labeling encodes, in a neat and efficient way, a generating function for the expressions of the remaining four eigenvectors that label the opposing face of the cube in terms of the coordinates represented by the chosen face. The main motivation behind this work is that when our matrix is a Lax matrix of a discrete integrable system, such generating functions can be interpreted as Lagrangians of the system, and a choice of a particular face corresponds to a choice of the direction of the motion.
In the framework of the focusing Nonlinear Schrodinger (NLS) equation we study numerically the nonlinear stage of the modulation instability (MI) of the condensate. As expected, the development of the MI leads to formation of integrable turbulence [V.E. Zakharov, Turbulence in integrable systems, Stud. in Appl. Math. 122, no. 3, 219-234, (2009)]. We study the time evolution of its major characteristics averaged across realizations of initial data - the condensate solution seeded by small random noise with fixed statistical properties. The measured quantities are: (1) wave-action spectrum and spatial correlation function, (2) the probability density function (PDF) of wave amplitudes and their momenta, and (3) kinetic and potential energies.
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