Dynamical symmetry breaking through AI: The dimer self-trapping transition

The nonlinear dimer obtained through the nonlinear Schrödinger equation has been a workhorse for the discovery the role nonlinearity plays in strongly interacting systems. While the analysis of the stationary states demonstrates the onset of a symmetry broken state for some degree of nonlinearity, the full dynamics maps the system into an effective [Formula: see text] model. In this later context, the self-trapping transition is an initial condition-dependent transfer of a classical particle over a barrier set by the nonlinear term. This transition that has been investigated analytically and mathematically is expressed through the hyperbolic limit of Jacobian elliptic functions. The aim of this work is to recapture this transition through the use of methods of Artificial Intelligence (AI). Specifically, we used a physics motivated machine learning model that is shown to be able to capture the original dynamic self-trapping transition and its dependence on initial conditions. Exploitation of this result in the case of the nondegenerate nonlinear dimer gives additional information on the more general dynamics and helps delineate linear from nonlinear localization. This work shows how AI methods may be embedded in physics and provide useful tools for discovery.

Kottler spacetime in isotropic static coordinates

The Kottler spacetime in isotropic coordinates is known where the metric is time-dependent. In this paper, the Kottler spacetime is given in isotropic static coordinates (i.e., the metric components are time-independent). The metric is found in terms of the Jacobian elliptic functions through coordinate transformations from the Schwarzschild-(anti-)de Sitter metric. In canonical coordinates, it is known that the unparameterized spatially projected null geodesics of the Kottler and Schwarzschild spacetimes coincide. We show that in isotropic static coordinates, the refractive indices of Kottler and Schwarzschild are not proportional, yielding spatially projected null geodesics that are different.

Inverses and n-uncial property of Jacobian elliptic functions

The inverses of Jacobi elliptic functions possess an apparently-non-crucial property: they provide almost-everywhere-conformal maps on a hemisphere onto a torus and so, onto a parallelogram. Thus, they produce map projections on the sphere generalizing the famous quincuncial projection of Charles S. Peirce. Besides providing a general practical definition of n-uncial map and proving that all the considered inverse elliptic functions are n-uncial, we give operative handy formulas to calculate these maps. To the best of our knowledge, these useful formulas have not been all together published before, except for Pierce projection. We look forward to their numerical implementation. By the way, we also classify the resulting map projections according the number of singularities.

The Dirichlet problem for rectangle and new identities for elliptic integrals and functions

In the paper, results of comparison of two different methods of exact solution of the Dirichlet problem for rectangle are presented, namely, method of conformal mapping and method of variables’ separation. By means of this procedure normal derivative of Green’s function for rectangular domain was expressed via Jacobian elliptic functions. Under approaching to rectangle’s boundaries these formulas give new representations of the Dirac delta function. Moreover in the framework of suggested ideology a number of identities for the complete elliptic integral of the first kind were obtained. These formulas may be applied to summation of both numerical and functional series; also they may be useful for analytic number theory.

A completely integrable case in the complex Lorenz equations

By application of the Lie theory of extended groups and for the parameter values σ=1/2, b=1, r1= e^2/2, r2=e/2, e arbitrary we prove that the system of the complex Lorenz equations is algebraically completely integrable. The respective general exact solution i$ expressed by means of Jacobian elliptic functions

EVOLUTION OF MODULATIONAL INSTABILITY IN TRAVELLING WAVE SOLUTION OF NON-LINEAR PARTIAL DIFFERENTIAL EQUATION

The Ritz variational method has been applied to the nonlinear partial differential equation to construct a model for travelling wave solution. The spatially periodic trial function was chosen in the form of combination of Jacobian Elliptic functions, with the dependence of its parameters

On solutions of the combined KdV-nKdV equation

The aim of this work is to deal with a new integrable nonlinear equation of wave propagation, the combined of the Korteweg-de vries equation and the negative order Korteweg-de vries equation (combined KdV-nKdV) equation, which was more recently proposed by Wazwaz. Upon using wave reduction variable, it turns out that the reduced combined KdV-nKdV equation is alike the reduced (3+1)-dimensional Jimbo Miwa (JM) equation, the reduced (3+1)-dimensional Potential Yu-Toda-Sasa-Fukuyama (PYTSF) equation and the reduced (3 + 1)¬dimensional generalized shallow water (GSW) equation in the trav¬elling wave. In fact, the four transformed equations belong to the same class of ordinary differential equation. With the benefit of a well known general solutions for the reduced equation, we show that sub¬jects to some scaling and change of parameters, a variety of families of solutions are constructed for the combined KdV-nKdV equation which can be expressed in terms of rational functions, exponential functions and periodic solutions of trigonometric functions and hyperbolic func¬tions. In addition to that the equation admits solitary waves, and double periodic waves in terms of special functions such as Jacobian elliptic functions and Weierstrass elliptic functions.