Search Results (49)

In this work, we consider a generalized form of the classical $(2+1)$-dimensional sine-Gordon system. The mathematical model considers a generalized reaction term, and the two-dimensional Laplacian includes the presence of space-fractional derivatives of the Riesz type with two different differentiation orders in general. The system is equipped with a conserved quantity that resembles the energy functional in the integer-order scenario. We propose a numerical model to approximate the solutions of the fractional sine-Gordon equation. A discretized form of the energy-like quantity is proposed, and we prove that it is conserved throughout the discrete time. Moreover, the analysis of consistency, stability, and convergence is rigorously carried out. The numerical model is implemented computationally, and some computer simulations are presented in this work. As a consequence of our simulations, we show that the discrete energy is approximately conserved throughout time, which coincides with the theoretical results. Full article

This study explores the (1+1)-dimensional Klein–Fock–Gordon equation, a distinct third-order nonlinear differential equation of significant theoretical interest. The Klein–Fock–Gordon equation (KFGE) plays a pivotal role in theoretical physics, modeling high-energy particles and providing a fundamental framework for simulating relativistic wave phenomena. To find the exact solution of the proposed model, for this purpose, we utilized two effective techniques, including the sine-Gordon equation method and a new extended direct algebraic method. The novelty of these approaches lies in the form of different solutions such as hyperbolic, trigonometric, and rational functions, and their graphical representations demonstrate the different form of solitons like kink solitons, bright solitons, dark solitons, and periodic waves. To illustrate the characteristics of these solutions, we provide two-dimensional, three-dimensional, and contour plots that visualize the magnitude of the (1+1)-dimensional Klein–Fock–Gordon equation. By selecting suitable values for physical parameters, we demonstrate the diversity of soliton structures and their behaviors. The results highlighted the effectiveness and versatility of the sine-Gordon equation method and a new extended direct algebraic method, providing analytical solutions that deepen our insight into the dynamics of nonlinear models. These results contribute to the advancement of soliton theory in nonlinear optics and mathematical physics. Full article

This paper presents a high-order structure-preserving difference scheme for the nonlinear space fractional sine-Gordon equation with damping, employing the triangular scalar auxiliary variable approach. The original equation is reformulated into an equivalent system that satisfies a modified energy conservation or dissipation law, significantly reducing the computational complexity of nonlinear terms. Temporal discretization is achieved using a second-order difference method, while spatial discretization utilizes a simple and easily implementable discrete approximation for the fractional Laplacian operator. The boundedness and convergence of the proposed numerical scheme under the maximum norm are rigorously analyzed, demonstrating its adherence to discrete energy conservation or dissipation laws. Numerical experiments validate the scheme’s effectiveness, structure-preserving properties, and capability for long-time simulations for both one- and two-dimensional problems. Additionally, the impact of the parameter $\epsilon$ on error dynamics is investigated. Full article

This paper introduces an enhanced Iterative Finite Difference (IFD) method for efficiently solving strongly nonlinear, time-dependent problems. Extending the original IFD framework for nonlinear ordinary differential equations, we generalize the approach to address nonlinear partial differential equations with time dependence. An improved strategy is developed to achieve high-order accuracy in space and time. A finite difference discretization is applied at each iteration, yielding a flexible and robust iterative scheme suitable for complex nonlinear equations, including the Sine-Gordon, Klein–Gordon, and generalized Sinh-Gordon equations. Numerical experiments confirm the method’s rapid convergence, high accuracy, and low computational cost. Full article

This review paper explores the Riccati-type pseudo-potential formulation applied to the quasi-integrable sine-Gordon, KdV, and NLS models. The proposed framework provides a unified methodology for analyzing quasi-integrability properties across various integrable systems, including deformations of the sine-Gordon, Bullough–Dodd, Toda, KdV, pKdV, NLS, and SUSY sine-Gordon models. Key findings include the emergence of infinite towers of anomalous conservation laws within the Riccati-type approach and the identification of exact non-local conservation laws in the linear formulations of deformed models. As modified integrable models play a crucial role in diverse fields of nonlinear physics—such as Bose–Einstein condensation, superconductivity, gravity models, optics, and soliton turbulence—these results may have far-reaching applications. Full article

In the present work, we explored the dynamics of single kinks, kink–anti-kink pairs and bound states in the prototypical fractional Klein–Gordon example of the sine-Gordon equation. In particular, we modified the order $\beta$ of the temporal derivative to that of a Caputo fractional type and found that, for $1<\beta <2$, this imposes a dissipative dynamical behavior on the coherent structures. We also examined the variation of a fractional Riesz order $\alpha$ on the spatial derivative. Here, depending on whether this order was below or above the harmonic value $\alpha =2$, we found, respectively, monotonically attracting kinks, or non-monotonic and potentially attracting or repelling kinks, with a saddle equilibrium separating the two. Finally, we also explored the interplay of the two derivatives, when both Caputo temporal and Riesz spatial derivatives are involved. Full article

This study presents a comprehensive investigation of cubic–quartic solitons within birefringent optical fibers, focusing on the effects of the Kerr law on the refractive index. The researchers have derived soliton solutions analytically using the sine-Gordon function technique. To validate their analytical results, the study employs the improved Adomian decomposition method, a numerical technique known for its efficiency and accuracy in solving nonlinear problems. This method effectively approximates solutions while minimizing computational errors, allowing for reliable numerical simulations that corroborate the analytical findings. The insights gained from this research contribute to a deeper understanding of the symmetry properties involved in nonlinear wave propagation in optical fibers. The study highlights the significant role of nonlinearities in shaping the behavior of waves within these systems. The use of proposed method not only serves as a checking mechanism for the sine-Gordon solutions but also illustrates its potential applicability to other nonlinear systems exhibiting complex symmetry behaviors. This versatility could lead to new exploration fronts in nonlinear optics and photonics, expanding the toolkit available for researchers in these rapidly evolving fields. Full article

This study employs a novel physics-informed neural network (PINN) approach, the standard explicit finite difference method (EFDM) and unconditionally positivity preserving FDM to tackle the one-dimensional Sine–Gordon equation (SGE). Two test problems with known analytical solutions are investigated to demonstrate the effectiveness of these techniques. While the three employed approaches demonstrate strong agreement, our analysis reveals that the EFDM results are in the best agreement with the analytical solutions. Given the consistent agreement between the numerical results from the EFDM, unconditionally positivity preserving FDM and PINN approach and the analytical solutions, all three methods are recommended as competitive options. The solution techniques employed in this study can be a valuable asset for present and future model developers engaged in various nonlinear physical wave phenomena, such as propagation of solitons in optical fibers. Full article

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 scalar field induces an outward flux of fermions and antifermions, leading to particle production and eventual flux propagation toward spatial infinity. Through numerical simulations, we observe that the fermion density exhibits quasiperiodic behavior, with partial recurrences of the bound state configurations after each oscillation period. However, the fermion wave functions do not remain localized, and over time, the density decreases as more particles escape the vicinity of the domain walls. Our results highlight that the sine-Gordon-like breather background, when coupled non-supersymmetrically to fermions, does not preserve integrability or stability, with the oscillations driving a continuous energy transfer into the fermionic modes. This study sheds light on the challenges of maintaining steady-state fermion solutions in time-dependent topological backgrounds and offers insights into particle production mechanisms in nonlinear dynamical systems with oscillating solitons. Full article

In this study, the energy control and asymptotic stability of the 1D sine-Gordon equation were investigated from the viewpoint of numerical approximation. An order reduction method was employed to transform the closed-loop system into an equivalent system, and an average-central finite difference scheme was constructed. This scheme is not only energy-preserving but also possesses uniform stability. The discrete multiplier method was utilized to obtain the uniformly asymptotic stability of the discrete systems. Moreover, to cope with the nonlinear term of the model, a discrete Wirtinger inequality suitable for our approximating scheme was established. Finally, several numerical experiments based on the eigenvalue distribution of the linearized approximation systems were conducted to demonstrate the effectiveness of the numerical approximating algorithm. Full article

In this paper, we consider the generalized sine-Gordon equation ${\psi }_{tx}=(1+a{\partial }_x^2)\sin \psi$ and the sinh-Poisson equation $u_{xx}+u_{yy}+\sigma \sinh u=0$, where a is a real parameter, and $\sigma$ is a positive parameter. Under different conditions, e.g., $a=0$, $a\ne 0$, and $\sigma >0$, the periods of the periodic wave solutions for the above two equations are discussed. By the transformation of variables, the generalized sine-Gordon equation and sinh-Poisson equations are reduced to planar dynamical systems whose first integral includes trigonometric terms and exponential terms, respectively. We successfully handle the trigonometric terms and exponential terms in the study of the monotonicity of the period function of periodic solutions. Full article

This paper establishes a novel technique, which is called the G-double-Laplace transform. This technique is an extension of the generalized Laplace transform. We study its properties with examples and various theorems related to the G-double-Laplace transform that have been addressed and proven. Finally, we apply the G-double-Laplace transform decomposition method to solve the nonlinear sine-Gordon and coupled sine-Gordon equations. This method is a combination of the G-double-Laplace transform and decomposition method. In addition, some examples are examined to establish the accuracy and effectiveness of this technique. Full article

The nonlinear Schrödinger (NLS) equation stands as a cornerstone model for exploring the intricate behavior of weakly nonlinear, quasi-monochromatic wave packets in dispersive media. Its reach extends across diverse physical domains, from surface gravity waves to the captivating realm of Bose–Einstein condensates. This article delves into the dual facets of the NLS equation: its capacity for modeling wave packet dynamics and its remarkable breadth of applications. We illuminate the derivation of the NLS equation through both heuristic and multiple-scale approaches, underscoring how distinct interpretations of physical variables and governing equations give rise to varied wave packet dynamics and tailored values for dispersive and nonlinear coefficients. To showcase its versatility, we present an overview of the NLS equation’s compelling applications in four research frontiers: nonlinear optics, surface gravity waves, superconductivity, and Bose–Einstein condensates. This exploration reveals the NLS equation as a powerful tool for unifying and understanding a vast spectrum of physical phenomena. Full article

Integrable quantum field theories can be regularized on the lattice while preserving integrability. The resulting theories on the lattice are integrable lattice models. A prototype of such a regularization is the correspondence between a sine-Gordon model and a six-vertex model on a light-cone lattice. We propose an integrable deformation of the light-cone lattice model such that in the continuum limit we obtain the $T\overline{T}$-deformed sine-Gordon model. Under this deformation, the cut-off momentum becomes energy dependent and the underlying Yang–Baxter integrability is preserved. Therefore, this deformation is integrable but non-local: similar to the $T\overline{T}$ deformation of quantum field theory. Full article

In this paper, the problems of estimating the parameters of partial differential equations from numerous observations in the vicinity of some reference points are considered. The paper is devoted to estimating the diffusion coefficient in the diffusion equation and the parameters of one-soliton solutions of nonlinear partial differential equations. When estimating the diffusion coefficient, it was necessary to construct an estimate of the second derivative based on inaccurate observations of the solution of the diffusion equation. This procedure required consideration of two reference points when determining the first and second partial derivatives of the solution of the diffusion equation. To analyse one-soliton solutions of partial differential equations, a series of techniques have been developed that allow one to estimate the parameters of the solution itself, but not its equation. These techniques are used to estimate the parameters of one-soliton solutions of the equations kdv, mkdv, Sine–Gordon, Burgers and nonlinear Schrodinger. All the considered estimates were tested during computational experiments. Full article