The Rise of Solitons in Sine-Gordon Field Theory: From Jacobi Amplitude to Gudermannian Function

Mondaini, Leonardo

doi:10.4236/jamp.2014.213141

Cited by 5 publications

(3 citation statements)

References 12 publications

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“…where ξ k ≡ ξ − ln k. Jacobi's amplitude function am (x, k) satisfies relations am(x, 0) = x, am (x, 1) = gd x, and can be considered as a generalization of the gudermannian function. In correspondence with this we note that x = x(χ) = am χ, which is the change of variable function in the case L C , appears as a generalization of the kink solution in the sine-Gordon model [74,75].…”

Section: Finite-gap Elliptic L-and D-familiessupporting

confidence: 66%

Position-dependent mass, finite-gap systems, and supersymmetry

Bravo

Plyushchay

2016

Phys. Rev. D

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The ordering problem in quantum systems with position-dependent mass (PDM) is treated by inclusion of the classically fictitious similarity transformation into the kinetic term. This provides a generation of supersymmetry with the first order supercharges from the kinetic term alone, while inclusion of the potential term allows also to generate nonlinear supersymmetry with higher order supercharges. A broad class of finite-gap systems with PDM is obtained by different reduction procedures, and general results on supersymmetry generation are applied to them. We show that elliptic finite-gap systems of Lamé and Darboux-Treibich-Verdier types can be obtained by reduction to Seiffert's spherical spiral and Bernoulli lemniscate in the presence of Calogero-like or harmonic oscillator potentials, or by angular momentum reduction of a free motion on some AdS 2 -related surfaces in the presence of Aharonov-Bohm flux. The limiting cases include the Higgs and Mathews-Lakshmanan oscillator models as well as a reflectionless model with PDM exploited recently in the discussion of cosmological inflationary scenarios.

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Section: Finite-gap Elliptic L-and D-familiessupporting

confidence: 66%

Position-dependent mass, finite-gap systems, and supersymmetry

Bravo

Plyushchay

2016

Phys. Rev. D

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“…Although there are other methods to obtain these translation-invariant solutions, e.g., the integral bifurcation method [39], some of these solutions, in particular the Weierstrass solutions of the Tzitzéica class of equations and the amplitude Jacobi solutions of the sine/sinh-Gordon equations cannot be obtained by the tanh method usually employed in the literature. Consequently, with a few exceptions in the case of the amplitude Jacobi solutions [33,34], their potential for realistic physical applications has been ignored in the past. As for the the Weierstrass solutions of the Tzitzéica class of equations, their potential in the area of optical solitons is still to be assessed [28,29].…”

Section: Discussionmentioning

confidence: 99%

Travelling-Wave Solutions for Wave Equations with Two Exponential Nonlinearities

Mancas

Rosu

Perez-Maldonado

2018

Zeitschrift Für Naturforschung A

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We use a simple method that leads to the integrals involved in obtaining the traveling-wave solutions of wave equations with one and two exponential nonlinearities. When the constant term in the integrand is zero, implicit solutions in terms of hypergeometric functions are obtained while when that term is nonzero, all the basic traveling-wave solutions of Liouville, Tzitzéica and their variants, as well as sine/sinh-Gordon equations with important applications in the phenomenology of nonlinear physics and dynamical systems are found through a detailed study of the corresponding elliptic equations.

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“…The use of the Gudermannian in calculus and applications is detailed in several contributions of some interest [13,22,25]: it is used in geodesy, in cartography to study Mercator map projection, (see for instance [24]), in soliton theory [20], in neural networks [29] and in mathematical statistics, where some probability density functions are introduced, taking inspiration to the sigmoid shape of the function [2,14].…”

Section: Introductionmentioning

confidence: 99%

A Structural Approach to Gudermannian Functions

Gambini,

Nicoletti,

Ritelli

2023

Results Math

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The Gudermannian function relates the circular angle to the hyperbolic one when their cosines are reciprocal. Whereas both such angles are halved areas of circular and hyperbolic sectors, it is natural to develop similar considerations within the study of a class of curves images of maps with constant areal speed. After a brief exposition of some use of the Gudermannian in applied sciences, we proceed to illustrate the class of curves, called Keplerian curves, which can be parametrised by a map $${{\textbf{m}}}= (\cos _{{{\textbf{m}}}}, \sin _{{{\textbf{m}}}})$$ m = ( cos m , sin m ) whose areal speed is 1. In the next Sections, after a detailed study of p-circular and hyperbolic Fermat curves $$\mathscr {F}_p$$ F p and $$\mathscr {F}^*_p$$ F p ∗ , we define the p-Gudermannian as the primitive of the derivative of the p-hyperbolic sine divided by the square of the p-hyperbolic cosine: all the analogues of the classical identities are proven. Having realised that such curves correspond to each other by means a homology, we extend our study to a wide class of Keplerian curves and their homologues; once again, defined the Gudermannian in an identical manner, all the analogues of classical identities subsist. Below, three examples are detailed. The last paragraph further extends this consideration, eliminating the hypothesis that the curves are parametrised by maps with areal speed 1. The Appendix illustrates integrating techniques for systems defining the Fermat curves and determining the inverse of their tangent function.

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The Rise of Solitons in Sine-Gordon Field Theory: From Jacobi Amplitude to Gudermannian Function

Abstract: We show how the famous soliton solution of the classical sine-Gordon field theory in (1 + 1)dimensions may be obtained as a particular case of a solution expressed in terms of the Jacobi amplitude, which is the inverse function of the incomplete elliptic integral of the first kind.

Cited by 5 publications

References 12 publications

Position-dependent mass, finite-gap systems, and supersymmetry

Position-dependent mass, finite-gap systems, and supersymmetry

Travelling-Wave Solutions for Wave Equations with Two Exponential Nonlinearities

A Structural Approach to Gudermannian Functions

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