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Register a new userDiscrete version of the Chazy class III equation
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We study the discretisation of the Chazy class III equation by two means: a discrete Painleve test, and the preservation of a two-parameter solution to the continuous equation. We get that way a best discretisation scheme.
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In the spirit of Kleins Erlangen Program, we investigate the geometric and algebraic structure of fundamental line complexes and the underlying privileged discrete integrable system for the minors of a matrix which constitute associated Plucker coordinates. Particular emphasis is put on the restriction to Lie circle geometry which is intimately related to the master dCKP equation of discrete integrable systems theory. The geometric interpretation, construction and integrability of fundamental line complexes in Mobius, Laguerre and hyperbolic geometry are discussed in detail. In the process, we encounter various avatars of classical and novel incidence theorems and associated cross- and multi-ratio identities for particular hypercomplex numbers. This leads to a discrete integrable equation which, in the context of Mobius geometry, governs novel doubly hexagonal circle patterns.
A demonstration of how the point symmetries of the Chazy Equation become nonlocal symmetries for the reduced equation is discussed. Moreover we construct an equivalent third-order differential equation which is related to the Chazy Equation under a generalized transformation, and find the point symmetries of the Chazy Equation are generalized symmetries for the new equation. With the use of singularity analysis and a simple coordinate transformation we construct a solution for the Chazy Equation which is given by a Right Painleve Series. The singularity analysis is applied to the new third-order equation and we find that it admits two solutions, one given by a Left Painleve Series and one given by a Right Painleve Series where the leading-order behaviors and the resonances are explicitly those of the Chazy Equation.
We point out a formal analogy between the Dirac equation in Majorana form and the discrete-velocity version of the Boltzmann kinetic equation. By a systematic analysis based on the theory of operator splitting, this analogy is shown to turn into a concrete and efficient computational method, providing a unified treatment of relativistic and non-relativistic quantum mechanics. This might have potentially far-reaching implications for both classical and quantum computing, because it shows that, by splitting time along the three spatial directions, quantum information (Dirac-Majorana wavefunction) propagates in space-time as a classical statistical process (Boltzmann distribution).
We show in this letter that the perturbed Burgers equation $u_t = 2uu_x + u_{xx} + epsilon ( 3 alpha_1 u^2 u_x + 3alpha_2 uu_{xx} + 3alpha_3 u_x^2 + alpha_4 u_{xxx} )$ is equivalent, through a near-identity transformation and up to order epsilon, to a linearizable equation if the condition $3alpha_1 - 3alpha_3 - 3/2 alpha_2 + 3/2 alpha_4 = 0$ is satisfied. In the case this condition is not fulfilled, a normal form for the equation under consideration is given. Then, to illustrate our results, we make a linearizability analysis of the equations governing the dynamics of a one-dimensional gas.
The main result of this paper is a discrete Lawson correspondence between discrete CMC surfaces in R^3 and discrete minimal surfaces in S^3. This is a correspondence between two discrete isothermic surfaces. We show that this correspondence is an isometry in the following sense: it preserves the metric coefficients introduced previously by Bobenko and Suris for isothermic nets. Exactly as in the smooth case, this is a correspondence between nets with the same Lax matrices, and the immersion formulas also coincide with the smooth case.
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