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

Leonardo Mondaini^1,2*

¹Department of Oncology, University of Alberta, Edmonton, Canada.
²Grupo de Física Teórica e Experimental, Departamento de Ciências Naturais, Universidade Federal do Estado do Rio de Janeiro, Rio de Janeiro, Brazil.

DOI: 10.4236/jamp.2014.213141                

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.

Keywords

Solitons, Sine-Gordon Field Theory, Elliptic Integrals, Jacobi Amplitude

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1]	Coleman, S. (1975) Quantum Sine-Gordon Equation as the Massive Thirring Model. Physical Review D, 11, 2088-2097. http://dx.doi.org/10.1103/PhysRevD.11.2088
[2]	Kosterlitz, J.M. (1974) The Critical Properties of the Two-Dimensional XY Model. Journal of Physics C: Solid State Physics, 7, 1046-1060. http://dx.doi.org/10.1088/0022-3719/7/6/005
[3]	Samuel, S. (1978) Grand Partition Function in Field Theory with Applications to Sine-Gordon Field Theory. Physical Review D, 18, 1916-1932. http://dx.doi.org/10.1103/PhysRevD.18.1916
[4]	Dauxois, T. and Peyrard, M. (2006) Physics of Solitons. Cambridge University Press, New York.
[5]	Mondaini, L. and Marino, E.C. (2005) Sine-Gordon/Coulomb Gas Soliton Correlation Functions and an Exact Evaluation of the Kosterlitz-Thouless Critical Exponent. Journal of Statistical Physics, 118, 767-779. http://dx.doi.org/10.1007/s10955-004-8828-y
[6]	Mondaini, L., Marino, E.C. and Schmidt, A.A. (2009) Vanishing Conductivity of Quantum Solitons in Polyacetylene. Journal of Physics A: Mathematical and Theoretical, 42, Article ID: 055401. http://dx.doi.org/10.1088/1751-8113/42/5/055401
[7]	Mondaini, L. (2012) Thermal Soliton Correlation Functions in Theories with a Z(N) Symmetry. Journal of Modern Physics, 3, 1776-1780. http://dx.doi.org/10.4236/jmp.2012.311221
[8]	Cervero, J.M. (1986) Unveiling the Solitons Mistery: The Jacobi Elliptic Functions. American Journal of Physics, 54, 35-38. http://dx.doi.org/10.1119/1.14767
[9]	Mondaini, L. (2012) Obtaining a Closed-Form Representation for the Dual Bosonic Thermal Green Function by Using Methods of Integration on the Complex Plane. Revista Brasileira de Ensino de Física, 34, 3305. http://dx.doi.org/10.1590/S1806-11172012000300005
[10]	Jackiw, R. (1977) Quantum Meaning of Classical Field Theory. Reviews of Modern Physics, 49, 681-706. http://dx.doi.org/10.1103/RevModPhys.49.681
[11]	Rajaraman, R. (1987) Solitons and Instantons: An Introduction to Solitons and Instantons in Quantum Field Theory. Elsevier, Amsterdam.
[12]	Gradshteyn, I.S. and Ryzhik, I.M. (2000) Table of Integrals, Series, and Products. Academic Press, San Diego.
[13]	Weisstein, E.W. Jacobi Amplitude. MathWorld—A Wolfram Web Resource. http://mathworld.wolfram.com/JacobiAmplitude.html