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Journal articles on the topic 'Jacobi elliptic functions'

Author: Grafiati

Published: 4 June 2021

Last updated: 25 July 2025

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1

Bawa, Parveen. "Elliptic Functions With A View Toward Jacobi Hyperbolic Functions." Global Journal For Research Analysis 3, no. 4 (2012): 66–68. http://dx.doi.org/10.15373/22778160/apr2014/21.

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2

Laville, Guy, and Ivan Ramadanoff. "Jacobi Elliptic Cliffordian Functions." Complex Variables, Theory and Application: An International Journal 47, no. 9 (2002): 787–802. http://dx.doi.org/10.1080/02781070290032243.

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3

HARDY, YORICK, W. H. STEEB, and R. STOOP. "JACOBI ELLIPTIC FUNCTIONS, NONLINEAR EVOLUTION EQUATIONS AND RECURSION." International Journal of Modern Physics C 11, no. 01 (2000): 27–31. http://dx.doi.org/10.1142/s0129183100000031.

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Jacobi elliptic functions play a central role in nonlinear evolution equations. We show how with one recursive call we can calculate the Jacobi elliptic functions sn, cn, and dn. We give an implementation using C++.

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4

Myung, Hyo Chul, and Dong Soo Lee. "Jacobi elliptic algebras of SO(3)." Glasgow Mathematical Journal 39, no. 1 (1997): 115–20. http://dx.doi.org/10.1017/s0017089500031980.

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5

Elías-Zúñiga, Alex, Luis Manuel Palacios-Pineda, Isaac H. Jiménez-Cedeño, Oscar Martínez-Romero, and Daniel Olvera Trejo. "He's frequency–amplitude formulation for nonlinear oscillators using Jacobi elliptic functions." Journal of Low Frequency Noise, Vibration and Active Control 39, no. 4 (2020): 1216–23. http://dx.doi.org/10.1177/1461348420972820.

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In this work, the Duffing’s type analytical frequency–amplitude relationship for nonlinear oscillators is derived by using Hés formulation and Jacobi elliptic functions. Comparison of the numerical results obtained from the derived analytical expression using Jacobi elliptic functions with respect to the exact ones is performed by considering weak and strong Duffing’s nonlinear oscillators.

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6

Johannessen, Kim. "A Nonlinear Differential Equation Related to the Jacobi Elliptic Functions." International Journal of Differential Equations 2012 (2012): 1–9. http://dx.doi.org/10.1155/2012/412569.

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A nonlinear differential equation for the polar angle of a point of an ellipse is derived. The solution of this differential equation can be expressed in terms of the Jacobi elliptic function dn(u,k). If the polar angle is extended to the complex plane, the Jacobi imaginary transformation properties and the dependence on the real and complex quarter periods can be described. From the differential equation of the polar angle, exact solutions of the Poisson Boltzmann and the sinh-Poisson equations are found in terms of the Jacobi elliptic functions.

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7

Khare, Avinash, and Uday Sukhatme. "Cyclic identities involving Jacobi elliptic functions." Journal of Mathematical Physics 43, no. 7 (2002): 3798–806. http://dx.doi.org/10.1063/1.1484541.

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8

Khare, Avinash, Arul Lakshminarayan, and Uday Sukhatme. "Local identities involving Jacobi elliptic functions." Pramana 62, no. 6 (2004): 1201–29. http://dx.doi.org/10.1007/bf02704435.

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9

Myung, Hyo Chul. "Note on the Jacobi Elliptic Functions." Journal of Algebra 200, no. 1 (1998): 134–40. http://dx.doi.org/10.1006/jabr.1997.7253.

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10

Zheng, Bin, and Qinghua Feng. "A New Approach for Solving Fractional Partial Differential Equations in the Sense of the Modified Riemann-Liouville Derivative." Mathematical Problems in Engineering 2014 (2014): 1–7. http://dx.doi.org/10.1155/2014/307371.

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Based on a fractional complex transformation, certain fractional partial differential equation in the sense of the modified Riemann-Liouville derivative is converted into another ordinary differential equation of integer order, and the exact solutions of the latter are assumed to be expressed in a polynomial in Jacobi elliptic functions including the Jacobi sine function, the Jacobi cosine function, and the Jacobi elliptic function of the third kind. The degree of the polynomial can be determined by the homogeneous balance principle. With the aid of mathematical software, a series of exact sol

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11

Gepreel, Khaled A. "Improved General Mapping Deformation Method for Nonlinear Partial Differential Equations in Mathematical Physics." Journal of Applied Mathematics 2013 (2013): 1–9. http://dx.doi.org/10.1155/2013/258396.

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We use the improved general mapping deformation method based on the generalized Jacobi elliptic functions expansion method to construct some of the generalized Jacobi elliptic solutions for some nonlinear partial differential equations in mathematical physics via the generalized nonlinear Klein-Gordon equation and the classical Boussinesq equations. As a result, some new generalized Jacobi elliptic function-like solutions are obtained by using this method. This method is more powerful to find the exact solutions for nonlinear partial differential equations.

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12

Gepreel, Khaled A., and A. R. Shehata. "Jacobi Elliptic Solutions for Nonlinear Differential Difference Equations in Mathematical Physics." Journal of Applied Mathematics 2012 (2012): 1–15. http://dx.doi.org/10.1155/2012/710375.

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We put a direct new method to construct the rational Jacobi elliptic solutions for nonlinear differential difference equations which may be called the rational Jacobi elliptic functions method. We use the rational Jacobi elliptic function method to construct many new exact solutions for some nonlinear differential difference equations in mathematical physics via the lattice equation and the discrete nonlinear Schrodinger equation with a saturable nonlinearity. The proposed method is more effective and powerful to obtain the exact solutions for nonlinear differential difference equations.

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13

Chen, Yong, Qi Wang, and Biao Lic. "Jacobi Elliptic Function Rational Expansion Method with Symbolic Computation to Construct New Doubly-periodic Solutions of Nonlinear Evolution Equations." Zeitschrift für Naturforschung A 59, no. 9 (2004): 529–36. http://dx.doi.org/10.1515/zna-2004-0901.

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A new Jacobi elliptic function rational expansion method is presented by means of a new general ansatz and is very powerful, with aid of symbolic computation, to uniformly construct more new exact doubly-periodic solutions in terms of rational form Jacobi elliptic function of nonlinear evolution equations (NLEEs). We choose a (2+1)-dimensional dispersive long wave equation to illustrate the method. As a result, we obtain the solutions found by most existing Jacobi elliptic function expansion methods and find other new and more general solutions at the same time. When the modulus of the Jacobi

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14

Brumberg, Victor A. "Theory compression with elliptic functions." Symposium - International Astronomical Union 172 (1996): 89–100. http://dx.doi.org/10.1017/s0074180900127172.

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Introduction of Jacobi elliptic functions in planetary, satellite and cometary problems of celestial mechanics is a transformation of variables to present the analytical theories of motion in the more compact form as compared with the traditional series in multiples of mean longitudes or mean anomalies.

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15

Yan, Zhenya. "Elliptic Function Solutions of (2+1)-dimensional Longwave – Shortwave Resonance Interaction Equation via a sinh-Gordon Expansion Method." Zeitschrift für Naturforschung A 59, no. 1-2 (2004): 23–28. http://dx.doi.org/10.1515/zna-2004-1-203.

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With the aid of symbolic computation, the sinh-Gordon equation expansion method is extended to seek Jacobi elliptic function solutions of (2+1)-dimensional long wave-short wave resonance interaction equation, which describe the long and short waves propagation at an angle to each other in a two-layer fluid. As a result, new Jacobi elliptic function solutions are obtained. When the modulus m of Jacobi elliptic functions approaches 1, we also deduce the singular oliton solutions; while when the modulus m→0, we get the trigonometric function solutions. - PACS: 02.30.Jr, 03.40.Kf

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16

Gepreel, Khaled A., and Amr M. S. Mahdy. "Algebraic computational methods for solving three nonlinear vital models fractional in mathematical physics." Open Physics 19, no. 1 (2021): 152–69. http://dx.doi.org/10.1515/phys-2021-0020.

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Abstract This research paper uses a direct algebraic computational scheme to construct the Jacobi elliptic solutions based on the conformal fractional derivatives for nonlinear partial fractional differential equations (NPFDEs). Three vital models in mathematical physics [the space-time fractional coupled Hirota Satsuma KdV equations, the space-time fractional symmetric regularized long wave (SRLW equation), and the space-time fractional coupled Sakharov–Kuznetsov (S–K) equations] are investigated through the direct algebraic method for more explanation of their novel characterizes. This appro

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17

Bravo Yuste, S. "The Rayleigh method with Jacobi elliptic functions." Journal of Sound and Vibration 133, no. 1 (1989): 180–84. http://dx.doi.org/10.1016/0022-460x(89)90993-0.

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18

Alimov, M. M. "Expression of Elliptic Functions in Terms of Jacobi Functions." Russian Mathematics 66, no. 7 (2022): 64–69. http://dx.doi.org/10.3103/s1066369x22070015.

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19

Gepreel, Khaled A., Taher A. Nofal, and Ali A. Al-Thobaiti. "The Modified Rational Jacobi Elliptic Functions Method for Nonlinear Differential Difference Equations." Journal of Applied Mathematics 2012 (2012): 1–30. http://dx.doi.org/10.1155/2012/427479.

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We modified the rational Jacobi elliptic functions method to construct some new exact solutions for nonlinear differential difference equations in mathematical physics via the lattice equation, the discrete nonlinear Schrodinger equation with a saturable nonlinearity, the discrete nonlinear Klein-Gordon equation, and the quintic discrete nonlinear Schrodinger equation. Some new types of the Jacobi elliptic solutions are obtained for some nonlinear differential difference equations in mathematical physics. The proposed method is more effective and powerful to obtain the exact solutions for nonl

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20

Conway, John T. "Indefinite integrals of incomplete elliptic integrals from Jacobi elliptic functions." Integral Transforms and Special Functions 28, no. 6 (2017): 443–59. http://dx.doi.org/10.1080/10652469.2017.1304938.

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21

Wang, Bo, and Guiqin Jin. "Solving a Class of Nonlinear Evolution Equations Using Jacobi Elliptic Functions." Journal of Physics: Conference Series 2381, no. 1 (2022): 012038. http://dx.doi.org/10.1088/1742-6596/2381/1/012038.

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Abstract Nonlinear evolution equations play a crucial role in many applied scientific disciplines. Finding the exact solutions to such equations can contribute to a wide variety of applied disciplines. To be able to calculate exact periodic solutions of nonlinear evolution equations, this paper discusses the expansion of the Jacobi elliptic function and finds the proper expansion of the function. Additionally, by analyzing the applicable conditions of the function according to its specific properties, we find the exact periodic solutions to nonlinear evolution equations with constant coefficie

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22

Xiao, Yafeng, Haili Xue, and Hongqing Zhang. "A New Extended Jacobi Elliptic Function Expansion Method and Its Application to the Generalized Shallow Water Wave Equation." Journal of Applied Mathematics 2012 (2012): 1–21. http://dx.doi.org/10.1155/2012/896748.

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With the aid of symbolic computation, a new extended Jacobi elliptic function expansion method is presented by means of a new ansatz, in which periodic solutions of nonlinear evolution equations, which can be expressed as a finite Laurent series of some 12 Jacobi elliptic functions, are very effective to uniformly construct more new exact periodic solutions in terms of Jacobi elliptic function solutions of nonlinear partial differential equations. As an application of the method, we choose the generalized shallow water wave (GSWW) equation to illustrate the method. As a result, we can successf

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23

BAI, CHENG-JIE, and HONG ZHAO. "NEW SOLITARY WAVE AND JACOBI PERIODIC WAVE EXCITATIONS IN (2+1)-DIMENSIONAL BOITI–LEON–MANNA–PEMPINELLI SYSTEM." International Journal of Modern Physics B 22, no. 15 (2008): 2407–20. http://dx.doi.org/10.1142/s021797920803954x.

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By means of the multilinear variable separation approach, a general variable separation solution of the Boiti–Leon–Manna–Pempinelli equation is derived. Based on the general solution, some new types of localized structures — compacton and Jacobi periodic wave excitations are obtained by introducing appropriate lower-dimensional piecewise smooth functions and Jacobi elliptic functions.

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24

Zhao, Yun-Mei. "F-Expansion Method and Its Application for Finding New Exact Solutions to the Kudryashov-Sinelshchikov Equation." Journal of Applied Mathematics 2013 (2013): 1–7. http://dx.doi.org/10.1155/2013/895760.

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Based on theF-expansion method, and the extended version ofF-expansion method, we investigate the exact solutions of the Kudryashov-Sinelshchikov equation. With the aid of Maple, more exact solutions expressed by Jacobi elliptic function are obtained. When the modulus m of Jacobi elliptic function is driven to the limits 1 and 0, some exact solutions expressed by hyperbolic function solutions and trigonometric functions can also be obtained.

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25

Zayed, Elsayed M. E., Mahmoud El-Horbaty, Mohamed E. M. Alngar, and Mona El-Shater. "Dispersive Optical Solitons for Stochastic Fokas-Lenells Equation With Multiplicative White Noise." Eng 3, no. 4 (2022): 523–40. http://dx.doi.org/10.3390/eng3040037.

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For the first time, we study the Fokas–Lenells equation in polarization preserving fibers with multiplicative white noise in Itô sense. Four integration algorithms are applied, namely, the method of modified simple equation (MMSE), the method of sine-cosine (MSC), the method of Jacobi elliptic equation (MJEE) and ansatze involving hyperbolic functions. Jacobi-elliptic function solutions, bright, dark, singular, combo dark-bright and combo bright-dark solitons are presented.

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26

Khan, Muhammad Ishfaq, Waqas Ali Faridi, Muhammad Amin Sadiq Murad, Mujahid Iqbal, Ratbay Myrzakulov, and Zhanar Umurzakhova. "The Formation and Propagation of Soliton Wave Profiles for the Shynaray-IIa Equation." Acta Mechanica et Automatica 19, no. 1 (2025): 136–47. https://doi.org/10.2478/ama-2025-0016.

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Abstract This study examines a novel use of the Jacobi elliptic function expansion method to solve the Shynaray-IIA equation, a significant nonlinear partial differential equation that arises in optical fiber, plasma physics, surface symmetry geometry, and many other mathematical physics domains. This kind of solution has never been attained in research prior to this study. Numerous properties of a particular class of solutions, called the Jacobi elliptic functions, make them useful for the analytical solution of a wide range of nonlinear problems. Using this powerful method, we derive a set o

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27

Lekner, John. "Four solutions of a two-cylinder electrostatic problem, and identities resulting from their equivalence." Quarterly Journal of Mechanics and Applied Mathematics 73, no. 3 (2020): 251–60. http://dx.doi.org/10.1093/qjmam/hbaa010.

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Summary Four distinct solutions exist for the potential distribution around two equal circular parallel conducting cylinders, charged to the same potential. Their equivalence is demonstrated, and the resulting analytical identities are discussed. The identities relate the Jacobi elliptic function $sn$, the Jacobi theta functions $\theta _1 ,~\theta _2 $ and infinite series over trigonometric and hyperbolic functions.

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28

Dattoli, G., C. Chiccoli, S. Lorenzutta, G. Maino, M. Richetta, and A. Torre. "Generating functions of multivariable generalized Bessel functions and Jacobi‐elliptic functions." Journal of Mathematical Physics 33, no. 1 (1992): 25–36. http://dx.doi.org/10.1063/1.529959.

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29

Ulutas, Esma, Mustafa Inc, Dumitru Baleanu, and Sunil Kumar. "Bright, dark, and singular optical soliton solutions for perturbed Gerdjikov-Ivanov equation." Thermal Science 25, Spec. issue 2 (2021): 151–56. http://dx.doi.org/10.2298/tsci21s2151u.

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This study consider Gerdjikov-Ivanov equation where the perturbation terms appear with full non-linearity. The Jacobi elliptic function ansatz method is implemented to obtain exact solutions of this equation that models pulse dynamics in optical fibers. It is retrieved some bright, dark optical and singular solitons profile in the limiting cases of the Jacobi elliptic functions. The constraint conditions depending on the parameters for the existence of solitons are also presented.

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30

Abdel Kader, A. H., and M. S. Abdel Latif. "New soliton solutions of the CH–DP equation using Lie symmetry method." Modern Physics Letters B 32, no. 20 (2018): 1850234. http://dx.doi.org/10.1142/s0217984918502342.

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In this paper, using Lie symmetry method, we obtain some new exact traveling wave solutions of the Camassa–Holm–Degasperis–Procesi (CH–DP) equation. Some new bright and dark soliton solutions are obtained. Also, some new doubly periodic solutions in the form of Jacobi elliptic functions and Weierstrass elliptic functions are obtained.

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31

Alimov, Mars Myasumovich. "The expression of elliptic functions in terms of Jacobi functions." Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika, no. 7 (2022): 73–78. http://dx.doi.org/10.26907/0021-3446-2022-7-73-78.

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32

Chouikha, A. Raouf. "On Properties of Elliptic Jacobi Functions and Applications." Journal of Nonlinear Mathematical Physics 12, no. 2 (2005): 162–69. http://dx.doi.org/10.2991/jnmp.2005.12.2.2.

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33

Cervero, José M. "Unveiling the solitons mystery: The Jacobi elliptic functions." American Journal of Physics 54, no. 1 (1986): 35–38. http://dx.doi.org/10.1119/1.14767.

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34

Tchier, Fairouz, Ebru Cavlak Aslan, and Mustafa Inc. "Optical solitons for cascaded system: Jacobi elliptic functions." Journal of Modern Optics 63, no. 21 (2016): 2298–307. http://dx.doi.org/10.1080/09500340.2016.1198502.

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35

Khare, Avinash, and Uday Sukhatme. "Connecting Jacobi elliptic functions with different modulus parameters." Pramana 63, no. 5 (2004): 921–36. http://dx.doi.org/10.1007/bf02704331.

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36

Khare, Avinash, Arul Lakshminarayan, and Uday Sukhatme. "Cyclic identities for Jacobi elliptic and related functions." Journal of Mathematical Physics 44, no. 4 (2003): 1822–41. http://dx.doi.org/10.1063/1.1560856.

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37

Ramadanoff, I. "On the So-Called “Jacobi Elliptic Cliffordian Functions”." Advances in Applied Clifford Algebras 19, no. 3-4 (2009): 861–67. http://dx.doi.org/10.1007/s00006-009-0190-3.

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38

Popivanov, Petar, and Angela Slavova. "Traveling-Wave Solutions of Several Nonlinear Mathematical Physics Equations." Mathematics 13, no. 6 (2025): 901. https://doi.org/10.3390/math13060901.

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This paper deals with several nonlinear partial differential equations (PDEs) of mathematical physics such as the concatenation model (perturbed concatenation model) from nonlinear fiber optics, the plane hydrodynamic jet theory, the Kadomtsev–Petviashvili PDE from hydrodynamic (soliton theory) and others. For the equation of nonlinear optics, we look for solutions of the form amplitude Q multiplied by eiΦ, Φ being linear. Then, Q is expressed as a quadratic polynomial of some elliptic function. Such types of solutions exist if some nonlinear algebraic system possesses a nontrivial solution. I

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39

Gandhi, Raja Rama, and Edigles Guedes. "On Andrica’s Conjecture, Cramér’s Conjecture, Gaps between Primes and Jacobi Theta Functions I." Bulletin of Mathematical Sciences and Applications 4 (May 2012): 16–19. http://dx.doi.org/10.18052/www.scipress.com/bmsa.4.16.

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40

Alurrfi, Khaled A. E., Ayad M. Shahoot, Mohamed O. M. Elmrid, Ali M. Almsiri, and Abdullah M. H. Arwiniya. "THE (G′/G) -EXPANSION METHOD FOR SOLVING A NONLINEAR PDE DESCRIBING THE NONLINEAR LOW-PASS ELECTRICAL LINES." EPH - International Journal of Applied Science 1, no. 4 (2015): 30–40. http://dx.doi.org/10.53555/eijas.v1i4.23.

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In this paper, we apply the (G′/G)-expansion method based on three auxiliary equations namely, the generalized Riccati equation, the Jacobi elliptic equation and the second order linear ordinary differential equation to find many new exact solutions of a nonlinear partial differential equation (PDE) describing the nonlinear low-pass electrical lines. The given nonlinear PDE has been derived and can be reduced to a nonlinear ordinary differential equation (ODE) using a simple transformation. Solitons wave solutions, periodic functions solutions, rational functions solutions and Jacobi elliptic

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41

Shqair, Mohammed, Mohammed Alabedalhadi, Shrideh Al-Omari, and Mohammed Al-Smadi. "Abundant Exact Travelling Wave Solutions for a Fractional Massive Thirring Model Using Extended Jacobi Elliptic Function Method." Fractal and Fractional 6, no. 5 (2022): 252. http://dx.doi.org/10.3390/fractalfract6050252.

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The fractional massive Thirring model is a coupled system of nonlinear PDEs emerging in the study of the complex ultrashort pulse propagation analysis of nonlinear wave functions. This article considers the NFMT model in terms of a modified Riemann–Liouville fractional derivative. The novel travelling wave solutions of the considered model are investigated by employing an effective analytic approach based on a complex fractional transformation and Jacobi elliptic functions. The extended Jacobi elliptic function method is a systematic tool for restoring many of the well-known results of complex

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42

Guha-Roy, Chanchal. "ON EXPLICIT SOLUTIONS OF A COUPLED KdV-mKdV EQUATION." International Journal of Modern Physics B 03, no. 06 (1989): 871–75. http://dx.doi.org/10.1142/s0217979289000646.

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43

Guedes, Edigles, and K. Raja Rama Gandhi. "On an Infinite Product for the Ratio of Consecutive Prime Numbers." Bulletin of Mathematical Sciences and Applications 6 (November 2013): 33–39. http://dx.doi.org/10.18052/www.scipress.com/bmsa.6.33.

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44

Sekatskii, Sergey. "On the Applications of the Generalized Littlewood Theorem Concerning Integrals of the Logarithm of Analytical Functions to Elliptic Functions." Axioms 12, no. 6 (2023): 595. http://dx.doi.org/10.3390/axioms12060595.

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Recently, we have established and used the generalized Littlewood theorem concerning contour integrals of the logarithm of analytical function to obtain new criteria equivalent to the Riemann hypothesis. Later, the same theorem was applied to calculate certain infinite sums and study the properties of zeroes of a few analytical functions. In this study, we apply this approach to elliptic functions of Jacobi and Weierstrass. Numerous sums over inverse powers of zeroes and poles are calculated, including some results for the Jacobi elliptic functions sn(z, k) and others understood as functions o

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45

Salas, Alvaro H., Lorenzo J. Martinez H, and David L. Ocampo R. "New Solutions for the Generalized BBM Equation in terms of Jacobi and Weierstrass Elliptic Functions." Abstract and Applied Analysis 2021 (April 22, 2021): 1–10. http://dx.doi.org/10.1155/2021/5513266.

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The Jacobi elliptic function method is applied to solve the generalized Benjamin-Bona-Mahony equation (BBM). Periodic and soliton solutions are formally derived in a general form. Some particular cases are considered. A power series method is also applied in some particular cases. Some solutions are expressed in terms of the Weierstrass elliptic function.

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46

LUO, LIN, and ENGUI FAN. "VARIABLE SEPARATION AND ALGEBRAIC-GEOMETRIC SOLUTIONS OF MODIFIED TODA LATTICE EQUATION." Modern Physics Letters B 24, no. 19 (2010): 2041–55. http://dx.doi.org/10.1142/s0217984910024420.

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In this paper, we derive a modified Toda lattice hierarchy by resorting to the Lenard operator pairs. The hyper-elliptic curve and Abel–Jacobi coordinates are then introduced to linearize the associated flow, from which some algebraic-geometric solutions of the modified Toda lattice are explicitly constructed in terms of Riemann theta functions by using Jacobi inversion technique.

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47

Korenkov, Mykola, and Yurii Kharkevych. "On the Asymptotics and Distribution of Values of the Jacobi Theta Functions and the Estimate of the Type of the Weierstrass Sigma Functions." Axioms 11, no. 1 (2021): 12. http://dx.doi.org/10.3390/axioms11010012.

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A refined asymptotics of the Jacobi theta functions and their logarithmic derivatives have been received. The asymptotics of the Nevanlinna characteristics of the indicated functions and the arbitrary elliptic function have been found. The estimation of the type of the Weierstrass sigma functions has been given.

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48

TEH, ROSY, PEI-YEN TAN, and KHAI-MING WONG. "JACOBI ELLIPTIC MONOPOLE–ANTIMONOPOLE PAIR SOLUTIONS." International Journal of Modern Physics A 27, no. 26 (2012): 1250148. http://dx.doi.org/10.1142/s0217751x12501485.

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We present new classical generalized Jacobi elliptic one monopole–antimonopole pair (MAP) solutions of the SU(2) Yang–Mills–Higgs theory with the Higgs field in the adjoint representation. These generalized 1-MAP solutions are solved with θ-winding number m = 1 and ϕ-winding number n = 1, 2, 3,…,6. Similar to the generalized Jacobi elliptic one monopole solutions, these generalized 1-MAP solutions are solved by generalizing the large distance behavior of the solutions to the Jacobi elliptic functions and solving the second order equations of motion numerically when the Higgs potential is vanis

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49

Salas, Alvaro H., S. A. El-Tantawy, and Lorenzo J. H. Martínez. "Approximate Analytical and Numeric Solutions to a Forced Damped Gardner Equation." Scientific World Journal 2022 (May 11, 2022): 1–10. http://dx.doi.org/10.1155/2022/3240918.

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In this paper, some exact traveling wave solutions to the integrable Gardner equation are reported. The ansatz method is devoted for deriving some exact solutions in terms of Jacobi and Weierstrass elliptic functions. The obtained analytic solutions recover the solitary waves, shock waves, and cnoidal waves. Also, the relation between the Jacobi and Weierstrass elliptic functions is obtained. In the second part of this work, we derive some approximate analytic and numeric solutions to the nonintegrable forced damped Gardner equation. For the approximate analytic solutions, the ansatz method is

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50

TEH, ROSY, KHAI-MING WONG, and KOK-GENG LIM. "GENERALIZED JACOBI ELLIPTIC ONE-MONOPOLE: TYPE A." International Journal of Modern Physics A 25, no. 31 (2010): 5731–46. http://dx.doi.org/10.1142/s0217751x10051062.

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Abstract:

We present a new classical generalized one-monopole solution of the SU(2) Yang–Mills–Higgs theory with the Higgs field in the adjoint representation. We show that this generalized solution with θ-winding number m = 1 and ϕ-winding number n = 1 is an axially symmetric Jacobi elliptic generalization of the 't Hooft–Polyakov one-monopole. We construct this axially symmetric one-monopole solution by generalizing the large distance asymptotic solution of the 't Hooft–Polyakov one-monopole to the Jacobi elliptic functions and solving the second-order equations of motion numerically when the Higgs po

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