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Journal articles on the topic 'Sine-Gordon Type System'

Author: Grafiati

Published: 4 June 2025

Last updated: 1 August 2025

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1

S., Manna M. Sabawi and Y. Sabawi. "A Numerical Solution for Sine-Gordon Type System." System, Tikrit Journal of Pure Science 15, no. 3 (2010): 106–13. https://doi.org/10.5281/zenodo.3370232.

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Abstract  A numerical solution for Sine-Gordon type system was done by the use of two finite difference schemes, the first is the explicit scheme and the second is the Crank-Nicholson scheme. A comparison between the two schemes showed that the explicit scheme is easier and has faster convergence than the Crank-Nicholson scheme which is more accurate. The MATLAB environment  was used for the numerical computations.

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2

Khasanov, A. B., Kh N. Normurodov, and T. G. Khasanov. "Integration of a nonlinear sine-Gordon–Liouville-type equation in the class of periodic infinite-gap functions." Ukrains’kyi Matematychnyi Zhurnal 76, no. 8 (2024): 1217–34. http://dx.doi.org/10.3842/umzh.v76i8.7610.

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UDC 517.9 The method of inverse spectral problem is used to integrate a nonlinear sine-Gordon–Liouville-type equation in the class of periodic infinite-gap functions. The evolution of the spectral data for the periodic Dirac operator is introduced in which the coefficient of the Dirac operator is a solution of a nonlinear sine-Gordon–Liouville-type equation. The solvability of the Cauchy problemc is proved for an infinite system of Dubrovin differential equations in the class of three times continuously differentiable periodic infinite-gap functions. It is shown that the sum of a uniformly con

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3

Yildirim, Ozgur. "Weak solutions of unconditionally stable second-order difference schemes for nonlinear sine-Gordon systems." e-Journal of Analysis and Applied Mathematics 2024 (December 31, 2024): 47–70. https://doi.org/10.62780/ejaam/2024-005.

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This paper presents the existence and uniqueness of the weak solution for the nonlinear system of sine-Gordon equations which describes DNA dynamics. An unconditionally stable second order difference scheme generated by the unbounded operator A2 corresponding to the system of sine-Gordon equations is considered. Weak solutions are a more general type of solution to the system of sine-Gordon equations than classical solutions and are important in the case of low regularity conditions. The weak solvability is studied in the space of distributions using variational methods. A very efficient numer

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4

Khasanov, A. B., and Kh N. Normurodov. "Integration of a sine-Gordon type equation with an additional term in the class of periodic infinite-gap functions." Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika, no. 3 (April 9, 2024): 70–83. http://dx.doi.org/10.26907/0021-3446-2024-3-70-83.

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In this paper, the inverse spectral problem method is used to integrate a nonlinear sine-Gordon type equation with an additional term in the class of periodic infinite-gap functions. The solvability of the Cauchy problem for an infinite system of Dubrovin differential equations in the class of three times continuously differentiable periodic infinite-gap functions is proved. It is shown that the sum of a uniformly convergent functional series constructed by solving the system of Dubrovin equations and the first trace formula satisfies sine-Gordon-type equations with an additional term.

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5

REINISCH, G. "NONLINEAR KLEIN-GORDON SOLITON MECHANICS." International Journal of Modern Physics B 06, no. 21 (1992): 3395–440. http://dx.doi.org/10.1142/s021797929200150x.

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Nonlinear Klein-Gordon solitary waves — or “solitons” in a loose sense — in n+1 dimensions, driven by very general external fields which must only satisfy continuity — together with regularity conditions at the boundaries of the system, obey a quite simple equation of motion. This equation is the exact generalization to this dynamical system of infinite number of degrees of freedom — which may be conservative or not — of the second Newton’s law setting the basis of material point mechanics. In the restricted case of conservative nonlinear Klein-Gordon systems, where the external driving force

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6

Ivanov, B. A., and E. V. Tartakovskaya. "Phonon relaxation of kink-type solitons in quasi-one-dimensional ordered systems." Soviet Journal of Low Temperature Physics 12, no. 10 (1986): 613–18. https://doi.org/10.1063/10.0031592.

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The interaction of the order parameter of a one-dimensional nonlinear system with elastic excitations of a three-dimensional matrix is investigated. The kinetic coefficients of the “kink gas” are calculated in the flexible-chain and rigid-chain approximations for two models of the nonlinear system (sine–Gordon and Φ(4)). The contribution of phonon processes in the relaxation of the system to an equilibrium state is analyzed, and the temperature dependence of the kinetic coefficients of the kink gas is determined.

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7

Rout, Abhishek, and Brett Altschul. "Bound States and Particle Production by Breather-Type Background Field Configurations." Symmetry 16, no. 12 (2024): 1571. http://dx.doi.org/10.3390/sym16121571.

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We investigate the interaction of fermion fields with oscillating domain walls, inspired by breather-type solutions of the sine-Gordon equation, a nonlinear system of fundamental importance. Our study focuses on the fermionic bound states and particle production induced by a time-dependent scalar background field. The fermions couple to two domain walls undergoing harmonic motion, and we explore the resulting dynamics of the fermionic wave functions. We demonstrate that while fermions initially form bound states around the domain walls, the energy provided by the oscillatory motion of the scal

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8

Ma, Hong-Cai, and Sen Yue Lou. "Solutions Generated from the Symmetry Group of the (2 + 1)-Dimensional Sine-Gordon System." Zeitschrift für Naturforschung A 60, no. 4 (2005): 229–36. http://dx.doi.org/10.1515/zna-2005-0403.

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9

Jha, Navnit, Venu Gopal, and Bhagat Singh. "Geometric grid network and third-order compact scheme for solving nonlinear variable coefficients 3D elliptic PDEs." International Journal of Modeling, Simulation, and Scientific Computing 09, no. 06 (2018): 1850053. http://dx.doi.org/10.1142/s1793962318500538.

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By using nonuniform (geometric) grid network, a new high-order finite-difference compact scheme has been obtained for the numerical solution of three-space dimensions partial differential equations of elliptic type. Single cell discretization to the elliptic equation makes it easier to compute and exhibit stability of the numerical solutions. The monotone and irreducible property of the Jacobian matrix to the system of difference equations analyses the converging behavior of the numerical solution values. As an experiment, applications of the compact scheme to Schrödinger equations, sine-Gordo

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10

Selvaratnam, A. R., M. Vlieg-Hulstman, B. van-Brunt, and W. D. Halford. "On the solution of a class of second-order quasi-linear PDEs and the Gauss equation." ANZIAM Journal 42, no. 3 (2001): 312–23. http://dx.doi.org/10.1017/s1446181100011962.

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AbstractGauss' Theorema Egregium produces a partial differential equation which relates the Gaussian curvature K to components of the metric tensor and its derivatives. Well-known partial differential equations (PDEs) such as the Schrödinger equation and the sine-Gordon equation can be derived from Gauss' equation for specific choices of K and coördinate systems. In this paper we consider a class of Bäcklund Transformations which corresponds to coördinate transformations on surfaces with a given Gaussian curvature. These Bäcklund Transformations lead to the construction of solutions to certain

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11

Inc, Mustafa, Aliyu Isa Aliyu, Abdullahi Yusuf, and Dumitru Baleanu. "On the classification of conservation laws and soliton solutions of the long short-wave interaction system." Modern Physics Letters B 32, no. 18 (2018): 1850202. http://dx.doi.org/10.1142/s0217984918502020.

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In this paper, the classification of conservation laws (Cls) of the long short-wave interaction system (LSWS) which appears in fluid mechanics as well as plasma physics is implemented using two Cls theorems, namely, the multipliers approach and the new conservation theorem. The LSWS describes the interaction between one long longitudinal wave and one short transverse wave propagating in a generalized elastic medium. The zeroth-order multipliers and the nonlinear self-adjoint substitutions of the model are derived. Considering the fact that the new conservation theorem needs Lie point symmetrie

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12

Andrievsky, Boris, and Yury Orlov. "Numerical evaluation of sine-Gordon chain energy control via subdomains state feedback under quantizationand time sampling." Cybernetics and Physics, Volume 8, 2019, Number 1 (June 20, 2019): 18–28. http://dx.doi.org/10.35470/2226-4116-2019-8-1-18-28.

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The paper is devoted to the numerical performance evaluation of the speed-gradient algorithms, recently developed in (Orlov et al., 2018; Orlov et al., 2019) for controlling the energy of the sine-Gordon spatially distributed systems with several in-domain actuators. The influence of the level quantization of the state feedback control signal (possibly coupled to the time sampling) on the steady-state energy error and the closed loop system stability is investigated in the simulation study. The following types of quantization are taken into account: sampling-in-time control signal quantization

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13

Bak, S. M., and G. M. Kovtonyuk. "Well-posedness of the Cauchy problem for system of oscillators on 2D–lattice in weighted $l^2$-spaces." Matematychni Studii 56, no. 2 (2021): 176–84. http://dx.doi.org/10.30970/ms.56.2.176-184.

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We consider an infinite system of ordinary differential equations that describes the dynamics of an infinite system of  linearly coupled nonlinear oscillators on a two dimensional integer-valued lattice. It is assumed that each oscillator interacts linearly with its four nearest neighbors and the oscillators are at the rest at infinity. We study the initial value problem (the Cauchy problem) for such system. This system naturally can be considered as an operator-differential equation in the Hilbert, or even Banach, spaces of sequences. We note that $l^2$ is the simplest choic

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14

Iatkliang, Thitthita, Supaporn Kaewta, Nguyen Minh Tuan, and Sekson Sirisubtawee. "Novel Exact Traveling Wave Solutions for Nonlinear Wave Equations with Beta-Derivatives via the sine-Gordon Expansion Method." WSEAS TRANSACTIONS ON MATHEMATICS 22 (June 2, 2023): 432–50. http://dx.doi.org/10.37394/23206.2023.22.50.

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The main objectives of this research are to use the sine-Gordon expansion method (SGEM) along with the use of appropriate traveling transformations to extract new exact solitary wave solutions of the (2 + 1)- dimensional breaking soliton equation and the generalized Hirota-Satsuma coupled Korteweg de Vries (KdV) system equipped with beta partial derivatives. Using the chain rule, we convert the proposed nonlinear problems into nonlinear ordinary differential equations with integer orders. There is then no further demand for any normalization or discretization in the calculation process. The ex

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15

Sakhnovich, Alexander. "Construction of the Solution of the Inverse Spectral Problem for a System Depending Rationally on the Spectral Parameter, Borg–Marchenko-Type Theorem and Sine-Gordon Equation." Integral Equations and Operator Theory 69, no. 4 (2010): 567–600. http://dx.doi.org/10.1007/s00020-010-1843-2.

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16

DERKS, GIANNE, ARJEN DOELMAN, CHRISTOPHER J. K. KNIGHT, and HADI SUSANTO. "Pinned fluxons in a Josephson junction with a finite-length inhomogeneity." European Journal of Applied Mathematics 23, no. 2 (2011): 201–44. http://dx.doi.org/10.1017/s0956792511000301.

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We consider a Josephson junction system installed with a finite length inhomogeneity, either of micro-resistor or micro-resonator type. The system can be modelled by a sine-Gordon equation with a piecewise-constant function to represent the varying Josephson tunneling critical current. The existence of pinned fluxons depends on the length of the inhomogeneity, the variation in the Josephson tunneling critical current and the applied bias current. We establish that a system may either not be able to sustain a pinned fluxon, or – for instance by varying the length of the inhomogeneity – may exhi

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17

Yang, Xueping, and Chuanzhong Li. "Bäcklund transformations of Zn-sine-Gordon systems." Modern Physics Letters B 31, no. 17 (2017): 1750189. http://dx.doi.org/10.1142/s0217984917501895.

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In this paper, from the algebraic reductions from the Lie algebra [Formula: see text] to its commutative subalgebra [Formula: see text], we construct the general [Formula: see text]-sine-Gordon and [Formula: see text]-sinh-Gordon systems which contain many multi-component sine-Gordon type and sinh-Gordon type equations. Meanwhile, we give the Bäcklund transformations of the [Formula: see text]-sine-Gordon and [Formula: see text]-sinh-Gordon equations which can generate new solutions from seed solutions. To see the [Formula: see text]-systems clearly, we consider the [Formula: see text]-sine-Go

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18

Lu, Xiaowu, and Rudolf Schmid. "Symplectic integration of Sine–Gordon type systems." Mathematics and Computers in Simulation 50, no. 1-4 (1999): 255–63. http://dx.doi.org/10.1016/s0378-4754(99)00083-x.

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19

Sarafanov, G. F., and A. A. Utkin. "MODEL OF FORMATION AND PROPAGATION OF SLIP BANDS IN METALS." Problems of Strength and Plasticity 85, no. 1 (2023): 5–13. http://dx.doi.org/10.32326/1814-9146-2023-85-1-5-13.

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A theoretical study of the processes of localization of plastic deformation in metals has been carried out. Within the framework of the system of evolutionary equations for dislocation density, taking into account the multiplication and annihilation of dislocations, the possibility of a running solution for the slip strip is established. It is shown that the initial system has two equilibrium states. For the total dislocation density and dislocation charge normalized to a stationary homogeneous solution for dislocation density, these are the states (0,0) and (1,0) on the phase plane of the abo

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20

NAZAROV, V. N., K. YU SAMSONOV, and E. G. EKOMASOV. "ONE-DIMENSIONAL DYNAMICS OF THE DOMAIN BOUNDARY IN A SEVEN-LAYER FERROMAGNETIC STRUCTURE." Izvestia Ufimskogo Nauchnogo Tsentra RAN, no. 1 (March 31, 2023): 19–23. http://dx.doi.org/10.31040/2222-8349-2023-0-1-19-23.

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The dynamics of the domain boundary is considered using the example of a seven-layer ferromagnetic structure with three thin and four wide magnetic layers. The structure of the domain boundary is represented as a kink solution of the sine-Gordon equation. The equation of motion for magnetization was solved numerically using an explicit scheme. The discretization of the equation was carried out according to a standard five-point scheme of the "cross" type. The paper shows the features of the dynamics of the domain boundary in a multilayer magnetic system in the presence of thin magnetic layers

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21

Gesztesy, Fritz, and Helge Holden. "Dubrovin equations and integrable systems on hyperelliptic curves." MATHEMATICA SCANDINAVICA 91, no. 1 (2002): 91. http://dx.doi.org/10.7146/math.scand.a-14381.

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We introduce the most general version of Dubrovin-type equations for divisors on a hyperelliptic curve $\mathcal K_g$ of arbitrary genus $g\in\boldsymbol N$, and provide a new argument for linearizing the corresponding completely integrable flows. Detailed applications to completely integrable systems, including the KdV, AKNS, Toda, and the combined sine-Gordon and mKdV hierarchies, are made. These investigations uncover a new principle for $1+1$-dimensional integrable soliton equations in the sense that the Dubrovin equations, combined with appropriate trace formulas, encode all hierarchies o

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22

Zhang, Jian, Binlu Feng, Yufeng Zhang, and Long Ju. "Using Vector-Product Loop Algebra to Generate Integrable Systems." Axioms 12, no. 9 (2023): 840. http://dx.doi.org/10.3390/axioms12090840.

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A new three-dimensional Lie algebra and its loop algebra are proposed by us, whose commutator is a vector product. Based on this, a positive flow and a negative flow are obtained by introducing a new kind of spectral problem expressed by the vector product, which reduces to a generalized KdV equation, a generalized Schrödinger equation, a sine-Gordon equation, and a sinh-Gordon equation. Next, the well-known Tu scheme is generalized for generating isospectral integrable hierarchies and non-isospectral integrable hierarchies. It is important that we make use of the variational method to create

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23

Ekomasov, E. G., V. N. Nazarov, and K. Yu Samsonov. "Changing the Dynamic Parameters of Localized Breather and Soliton Waves in the Sine-Gordon Model with Extended Impurity, External Force, and Decay in the Autoresonance Mode." Nelineinaya Dinamika 18, no. 2 (2022): 217–29. http://dx.doi.org/10.20537/nd220205.

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Possibility of changing the dynamic parameters of localized breather and soliton waves for the sine-Gordon equation in the model with extended impurity, variable external force and dissipation was investigated using the autoresonance method. The model of ferromagnetic structure consisting of two wide identical layers separated by a thin layer with modified values of magnetic anisotropy parameter was taken as a basis. Frequency of external field is a linear function of time. The sine-Gordon equation (SGE) was solved numerically using the finite differences method with explicit scheme of integra

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24

Demskoi, Dmitry K. "The Lattice Sine-Gordon Equation as a Superposition Formula for an NLS-Type System." Symmetry, Integrability and Geometry: Methods and Applications, December 21, 2021. http://dx.doi.org/10.3842/sigma.2021.108.

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We treat the lattice sine-Gordon equation and two of its generalised symmetries as a compatible system. Elimination of shifts from the two symmetries of the lattice sine-Gordon equation yields an integrable NLS-type system. An auto-Bäcklund transformation and a superposition formula for the NLS-type system is obtained by elimination of shifts from the lattice sine-Gordon equation and its down-shifted version. We use the obtained formulae to calculate a superposition of two and three elementary solutions.

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25

"On a 2+1-dimensional integrable Ernst-type equation." Proceedings of the Royal Society of London. Series A: Mathematical and Physical Sciences 446, no. 1927 (1994): 381–98. http://dx.doi.org/10.1098/rspa.1994.0110.

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A strong 2+1-dimensional integrable extension of Ernst’s equation of general relativity is proposed. Its richness is demonstrated by means of various canonical dimensional reductions and specializations which lead to formal analogues of well-known 1+1- and 2+1-dimensional integrable systems such as the self-induced transparency equations, the Konopelchenko-Rogers equations, a 2+1- dimensional Darboux system descriptive of conjugate coordinate systems, a single 2+1-dimensional sine-Gordon equation and the equations representing its Bäcklund transformation. A Darboux-Levi-type transformation is

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26

Ekomasov, Evgenii, Kirill Samsonov, Azamat Gumerov, and Roman Kudryavtsev. "Nonlinear waves of the sine-Gordon equation in the model with three attracting impurities." Izvestiya VUZ. Applied Nonlinear Dynamics, October 5, 2022. http://dx.doi.org/10.18500/0869-6632-003011.

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Purpose of this work is to use analytical and numerical methods to consider the problem of the structure and dynamics of coupled localized nonlinear waves in the sine-Gordon model with impurities (or spatial inhomogeneity of the periodic potential). Methods. Using the analytical method of collective coordinates for the case of the arbitrary number the same point impurities on the same distance each other, differential equation system was got for localized waves amplitudes as the functions on time. We used the finite difference method with explicit scheme for the numerical solution of the modif

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27

Akkılıc, Ayse Nur, Tukur Abdulkadir Sulaiman, and Hasan Bulut. "Applications of the extended rational sine-cosine and sinh-cosh techniques to some nonlinear complex models arising in mathematical physics." Applied Mathematics and Nonlinear Sciences, April 6, 2021. http://dx.doi.org/10.2478/amns.2021.1.00021.

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Abstract This study presents the applications of the extended rational sine-cosine/sinh-cosh schemes to the Klein-Gordon-Zakharov equations and the (2+1)-dimensional Maccari system. Various wave solutions such as singular periodic, periodic wave, topological, topological kink-type, dark and singular soliton solutions are successfully revealed. To display the physical features of the reported solutions, we use some appropriate choice of parameters in plotting the 3D, 2D, and contour graphs of some attained solutions.

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28

Blas, H., M. Cerna Maguiña, and L. F. dos Santos. "Modified AKNS model, Riccati-type pseudo-potential approach and infinite towers of quasi-conservation laws." International Journal of Modern Physics B, April 18, 2022. http://dx.doi.org/10.1142/s0217979222500709.

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In this paper, a dual Riccati-type pseudo-potential formulation is introduced for a modified AKNS system (MAKNS) and infinite towers of novel anomalous conservation laws are uncovered. In addition, infinite towers of exact nonlocal conservation laws are uncovered in a linear formulation of the system. It is shown that certain modifications of the nonlinear Schrödinger model (MNLS) can be obtained through a reduction process starting from the MAKNS model. So, the novel infinite sets of quasi-conservation laws and related anomalous charges are constructed by an unified and rigorous approach base

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29

Macías-Díaz, Jorge E., and Anastasios Bountis. "Nonlinear Supratransmission in Quartic Hamiltonian Lattices With Globally Interacting Particles and On-Site Potentials." Journal of Computational and Nonlinear Dynamics 16, no. 2 (2020). http://dx.doi.org/10.1115/1.4048714.

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Abstract We investigate a family of one-dimensional (1D) Hamiltonian semi-infinite particle lattices whose interactions involve exclusively terms of fourth order in the potential. Our aim is to examine their distinct role in the dynamics, in the absence of quadratic (harmonic) interactions, which are typically included in most studies, as they are known to play an important role in many physical phenomena. We also include in our potentials on-site terms of the sine-Gordon type, which are also considered in many studies in connection with localization effects. Our 1D lattices are subjected to s

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30

Sahoo, Mrutyunjaya, and Snehashish Chakraverty. "Analytical Soliton Solutions, Bifurcation and Chaotic Behaviour of the Geophysical Boussinesq Equation." Journal of Computational and Nonlinear Dynamics, April 11, 2025, 1–19. https://doi.org/10.1115/1.4068424.

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Abstract This article investigates the traveling wave solution for a Geophysical Boussinesq-type equation, which serves as a model for equatorial tsunami waves. By employing a combination of traveling wave transformation and the Sine-Gordon expansion method, the study introduces several new traveling wave solutions, including hyperbolic, trigonometric types, and bell-shaped forms, for this nonlinear model. Through appropriate parameter selection, the research utilizes two-dimensional (2D), three-dimensional (3D), and contour plots to efficiently depict the characteristics of specific solutions

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31

Messenger, Daniel A., and David M. Bortz. "Asymptotic consistency of the WSINDy algorithm in the limit of continuum data." IMA Journal of Numerical Analysis, December 12, 2024. https://doi.org/10.1093/imanum/drae086.

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Abstract In this work we study the asymptotic consistency of the weak-form sparse identification of nonlinear dynamics algorithm (WSINDy) in the identification of differential equations from noisy samples of solutions. We prove that the WSINDy estimator is unconditionally asymptotically consistent for a wide class of models that includes the Navier–Stokes, Kuramoto–Sivashinsky and Sine–Gordon equations. We thus provide a mathematically rigorous explanation for the observed robustness to noise of weak-form equation learning. Conversely, we also show that, in general, the WSINDy estimator is onl

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32

Caudrelier, Vincent, Matteo Stoppato, and Benoît Vicedo. "Classical Yang–Baxter Equation, Lagrangian Multiforms and Ultralocal Integrable Hierarchies." Communications in Mathematical Physics 405, no. 1 (2024). http://dx.doi.org/10.1007/s00220-023-04871-x.

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AbstractWe cast the classical Yang–Baxter equation (CYBE) in a variational context for the first time, by relating it to the theory of Lagrangian multiforms, a framework designed to capture integrability in a variational fashion. This provides a significant connection between Lagrangian multiforms and the CYBE, one of the most fundamental concepts of integrable systems. This is achieved by introducing a generating Lagrangian multiform which depends on a skew-symmetric classical r-matrix with spectral parameters. The multiform Euler–Lagrange equations produce a generating Lax equation which yie

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