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In this paper, a fourth-order energy-preserving time integrator is derived by improving the classical average vector field integrator. Combining the proposed novel time integrator with the Fourier pseudo-spectral spatial discretisation, we develop and analyze an energy-preserving fully discrete scheme for the sine-Gordon equation with periodic boundary conditions. Numerical results verify the energy preservation and the accuracy of the proposed fully discrete scheme. [ABSTRACT FROM AUTHOR]

The aim of this paper is to establish improved uniform error bounds under H α / 2 -norm (1 < α ≤ 2) for the long-time dynamics of the high-dimensional nonlinear space fractional sine-Gordon equation (NSFSGE) by a Lawson-type exponential integrator Fourier pseudo-spectral (LEI-FP) method. Firstly, a Lawson-type exponential integrator method is used to discretize the time direction. Then, the Fourier pseudo-spectral method is applied to discretize the space direction. We rigorously prove that the equation is energy conservation in a continuous state. Regularity compensation oscillation (RCO) technique is employed to strictly prove the improved uniform error bounds at O ε 2 τ in temporal semi-discretization and O h m + ε 2 τ in full-discretization up to the long-time T ε = T / ε 2 ( T > 0 fixed), respectively. To obtain the convergence order h m in space, we only need to directly prove it instead of proving that the numerical solution is H m + α / 2 -norm bounded as before. Complex NSFSGE and oscillatory NSFSGE are also discussed. This is the novel work to construct the improved uniform error bounds for the long-time dynamics of the high-dimensional nonlinear space fractional Klein-Gordon equation with non-polynomial nonlinearity. Finally, numerical examples in two-dimension and three-dimension are provided to confirm the improved error bounds, and we find drastically different evolving patterns between NSFSGE and the classical sine-Gordon equation. [ABSTRACT FROM AUTHOR]

This study analyzes the Hirota–Satsuma–Ito equation, which discusses the propagation of unidirectional shallow‐water waves and the interactions between two long waves with different dispersion forms. For the proposed equation, the sine‐Gordon expansion method has been considered. This method is derived from the sine‐Gordon equation. Different types of solutions, namely, bright, periodic, and dark‐bright soliton solutions, are derived. When these solutions are compared to other previously published research, to our knowledge, the study concludes that they are innovative, and this method was not applied to this equation. The validation of the obtained solutions is verified and plotted as three‐dimensional figures to comprehend physical phenomena. With the proper parameter values, distinct graphs are created to convey the physical representation of specific solutions. The results of this paper show that the method effectively improves a system's nonlinear dynamical behavior. This study will be useful to a wide range of engineers who specialize in engineering models. The findings show that the computational approach is successful, simple, and even applicable to complex systems. [ABSTRACT FROM AUTHOR]

In this paper, for quickly solving one‐ and two‐dimensional space‐fractional sine‐Gordon equations with distributed delay, we suggest several accelerated schemes of direct compact difference (DCD) methods. For one‐dimensional (1D) problems, with a function transformation, we construct an indirect compact difference (ICD) method, which requires less calculation cost than the corresponding DCD method, and prove under the appropriate conditions that ICD method has second‐order (resp. forth‐order) calculation accuracy in time (resp. space). By extending the argument for 1D case, we further obtain an ICD method for solving two‐dimensional (2D) problems and derive the similar convergence result. For ICD and DCD methods of 2D problems, we also give their alternative direction implicit (ADI) schemes. Moreover, for the fast implementations of ICD method of 1D problems and indirect ADI method of 2D problems, we further present their acceleration strategies. Finally, with a series of numerical experiments, the findings in this paper are further confirmed. [ABSTRACT FROM AUTHOR]

In this work, we obtained a new functional matrix using Clique-polynomials of complete graphs$\left({{K_n}} \right)$Kn with $n$n vertices and considered a new approach to solving the Sine–Gordon (SG) equation. The clique polynomial method transforms this equation into a system of algebraic equations. The solution will be drawn with the help of Newton Raphson’s method. Also, we employed the q-homotopy analysis transform method (q-HATM), which is the proper collision of the Laplace transform and the q-homotopy analysis method (q-HAM). To witness the reliability and accuracy of the considered schemes, some illustrations of the SG equation and double SG equation are considered. Here, the SG equation is solved easily and elegantly without using discretization or transformation of the equation by using the q-HATM. Also, in q-HATM, the presence of homotopy and axillary parameters allows us to have a large convergence region. The 3D surfaces of acquired solutions are drawn effectively. The tables of error analysis demonstrate the success of these methods. [ABSTRACT FROM AUTHOR]

In this study, the nonlinear two-dimensional Kadomtsev–Petviashvili-modified equal width (KP-MEW) equation is under investigation, which is described as a nonlinear wave model in weakly restoring forces and shallow water waves in the way of ferromagnetic, solitary waves in two dimensions with short amplitude and long wavelength and are also helpful in the investigation of various behaviors in nonlinear sciences. We successfully found the interesting and novel solitary wave results in bright solitons, kink solitons, dark solitons, anti-kink solitons, periodic singular solitons for nonlinear two-dimensional KP-MEW equation on the base of extension of simple equation technique under symbolic computational software Mathematica. The created solitons play an important role in nonlinear engineering and physics such as nonlinear optics, nonlinear dynamics, soliton wave theory, optical fiber and communication system. We are sure that these constructed results are novel and interesting which does not exist in the previous literature works. The graphical representation of constructed results is shown by 2D, 3D and contour graphs. The investigated research work proved that utilized technique is more concise, straightaway and efficient for study of other nonlinear partial differential equations. [ABSTRACT FROM AUTHOR]

We study harmonic maps from a subset of the complex plane to a subset of the hyperbolic plane. In Fotiadis and Daskaloyannis (Nonlinear Anal 214, 112546, 2022), harmonic maps are related to the sinh-Gordon equation and a Bäcklund transformation is introduced, which connects solutions of the sinh-Gordon and sine-Gordon equation. We develop this machinery in order to construct new harmonic maps to the hyperbolic plane. [ABSTRACT FROM AUTHOR]

This paper deals with the investigation of the newly obtained solitary solutions of perturbed optical metamaterial with Kerr law nonlinearity. Two novel analytical techniques, namely, the extended Riccati equation rational expansion (ERERE) method and the extended rational exp (− ( φ ′ φ) (ξ)) -expansion (EREE) methods are applied to develop new solitary solutions. The resultant wave solutions are periodic wave solution, singular kink, anti-peakon, dark, singular anti-kink, anti-kink, gray and singular soliton solutions. The detailed dynamics of the retrieved solutions are the most potent influence of the proposed methods which shows the efficacy and fruitfulness of both the methods. [ABSTRACT FROM AUTHOR]

Summary This paper contributes to the study of a new model called the q$$ \mathfrak{q} $$‐deformed equation or the q$$ \mathfrak{q} $$‐deformed tanh‐Gordon model. To understand physical systems with violated symmetries. We utilize the G′kG′+G+r$$ \left(\frac{{\mathcal{G}}^{\prime }}{k{\mathcal{G}}^{\prime }+\mathcal{G}+r}\right) $$‐expansion approach to solve the q$$ \mathfrak{q} $$‐deformed equation for specific parameter values. This method generates solutions that provide valuable insights into the system's dynamics and behavior. To verify the accuracy of our solutions, we also apply the finite difference technique to obtain numerical solutions to the q$$ \mathfrak{q} $$‐deformed equation. This dual approach ensures the reliability of our results. We present our findings using tables and graphics to enhance clarity and facilitate comparison between the analytical and numerical solutions. These visual aids help readers better understand the similarities and differences between the two approaches. The q$$ \mathfrak{q} $$‐deformation is significant as it models physical systems with nonstandard symmetry features, like extensivity, offering a more accurate representation of real‐world phenomena. The growing significance of this equation across various disciplines highlights its potential in advancing our understanding of complex physical systems. This paper contributes valuable knowledge about the q$$ \mathfrak{q} $$‐deformed equation, demonstrating its potential for accurately modeling physical systems with violated symmetries. Through both analytical and numerical techniques, we offer comprehensive solutions and validate their accuracy, with graphical representations enhancing the clarity and understanding of our results. This exploration of q$$ \mathfrak{q} $$‐deformation advances modeling techniques, providing a more precise depiction of real‐world processes with nonstandard symmetry features. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

The Cauchy matrix approach is developed to construct explicit solutions for some nonisospectral equations, including the nonisospectral Korteweg–de Vries (KdV) equation, the nonisospectral modified KdV equation, and the nonisospectral sine-Gordon equation. By means of a Sylvester equation, a set of scalar master functions { S (i,j)} is defined. We show how nonisospectral dispersion relations are introduced such that the evolutions of { S (i,j)} can be derived. Some identities of { S (i,j)} are employed in verifying solutions. Some explicit one-soliton and two-soliton solutions are illustrated together with analysis of their dynamics. [ABSTRACT FROM AUTHOR]

In this paper, we consider solutions for discrete systems arising from multi-term time-fractional diffusion equations. Using discrete sine transform techniques, we find that all-at-once systems of such equations have a structure similar to that of diagonal-plus-Toeplitz matrices. We establish a generalized circulant approximate inverse preconditioner for the all-at-once systems. Through a detailed analysis of the preconditioned matrices, we show that the spectrum of the obtained preconditioned matrices is clustered around one. We give some numerical examples to demonstrate the effectiveness of the proposed preconditioner. [ABSTRACT FROM AUTHOR]

We consider systems of N particles interacting on the unit circle through 2 π -periodic potentials. An example is the N -rotor problem that arises as the classical limit of coupled Josephson junctions and for various energies is known to have a wide range of behaviors such as global chaos and ergodicity, together with families of periodic solutions and transitions from order to chaos. We focus here on selected initial values for generalized systems in which the second order Euler–Lagrange equations reduce to first order equations, which we show by example can describe an ensemble of oscillators with associated emergent phenomena such as synchronization. A specific case is that of the Kuramoto model with well-known synchronization properties. We further demonstrate the relation of these models to field theories in 1 + 1 dimensions that allow static kink solitons satisfying first order Bogomolny equations, well-known in soliton physics, which correspond to the first order equations of the generalized N -rotor models. For the nonlinear pendulum, for example, the first order equations define the separatrix in the phase portrait of the system and correspond to kink solitons in the sine-Gordon equation. [ABSTRACT FROM AUTHOR]

We show that, given any static spacetime whose spatial slices are asymptotically Euclidean (or, more generally, asymptotically conic) manifolds modeled on the large end of the Schwarzschild exterior, there exist stationary solutions to the Klein–Gordon equation having Schwartz initial data. In fact, there exist infinitely many independent such solutions. The proof is a variational argument based on the long range nature of the effective potential. We give two sets of test functions which serve to verify the hypothesis of the variational argument. One set consists of cutoff versions of the hydrogen bound states and is used to prove the existence of eigenvalues near the hydrogen spectrum. [ABSTRACT FROM AUTHOR]

A combined deep machine learning (DML) and collocation based approach to solve the partial differential equations using artificial neural networks is proposed. The developed method is applied to solve problems governed by the Sine–Gordon equation (SGE), the scalar wave equation and elasto-dynamics. Two methods are studied: one is a space-time formulation and the other is a semi-discrete method based on an implicit Runge–Kutta (RK) time integration. The methodology is implemented using the Tensorflow framework and it is tested on several numerical examples. Based on the results, the relative normalized error was observed to be less than 5% in all cases. [ABSTRACT FROM AUTHOR]

In this paper, we use the sub-ODE method to analyze soliton solutions for the renowned nonlinear Klein-Gordon model (NLKGM). This method provides a variety of soliton solutions, including three positive solitons, three Jacobian elliptic function solutions, bright solitons, dark solitons, periodic solitons, rational solitons and hyperbolic function solutions. Applications for these solitons can be found in optical communication, fiber optic sensors, plasma physics, Bose-Einstein condensation and other areas. We also study some numerical solutions by using forward, backward, and central difference techniques. Moreover, we discuss variational integrators (VIs) using the projection technique for NLKGM. We develop a numerical solution for NLKGM using the discrete Euler lagrange equation, the Lagrangian and the Euler lagrange equation. At the end, in various dimensions, covering 3D, 2D, and contour, we will also plot several graphs for the obtained NLKGM solutions. A contour plot is a type of graphic representation that displays a three-dimensional surface on a two-dimensional plane by using contour lines. Each contour line in the plotted function represents one of the function's constant values, mapping the function's value across the plane. This model has been studied across multiple soliton solutions using various methods in the open literature, but this model for VIs and finite deference scheme (FDS) is the first time it has been studied. Within the various numerical techniques accessible for solving Hamiltonian systems, variational integrators distinguish themselves because of their symplectic quality. Here are some of the symplectic properties: symplectic orthogonality, energy conservation, area preservation, and structure preservation. [ABSTRACT FROM AUTHOR]

We establish the improved uniform error bounds on a Lawson-type exponential integrator Fourier pseudospectral (LEI-FP) method for the long-time dynamics of sine-Gordon equation where the amplitude of the initial data is O (ε) with 0 < ε ≪ 1 a dimensionless parameter up to the time at O (1 / ε 2) . The numerical scheme combines a Lawson-type exponential integrator in time with a Fourier pseudospectral method for spatial discretization, which is fully explicit and efficient in practical computation thanks to the fast Fourier transform. By separating the linear part from the sine function and employing the regularity compensation oscillation (RCO) technique which is introduced to deal with the polynomial nonlinearity by phase cancellation, we carry out the improved error bounds for the semi-discretization at O (ε 2 τ) instead of O (τ) according to classical error estimates and at O (h m + ε 2 τ) for the full-discretization up to the time T ε = T / ε 2 with T > 0 fixed. This is the first work to establish the improved uniform error bound for the long-time dynamics of the nonlinear Klein–Gordon equation with non-polynomial nonlinearity. The improved error bound is extended to an oscillatory sine-Gordon equation with O (ε 2) wavelength in time and O (ε - 2) wave speed, which indicates that the temporal error is independent of ε when the time step size is chosen as O (ε 2) . Finally, numerical examples are shown to confirm the improved error bounds and to demonstrate that they are sharp. [ABSTRACT FROM AUTHOR]

In this paper, we primarily focus on the nonlinear paraxial wave equation in Kerr media, a generalization of the nonlinear Schrödinger equation utilized to characterize the dynamics of optical beam propagation. By using three potent analytical methods, namely, the Sine-Gordon expansion method, the functional variable method, and the Bernoulli (G′/G)-expansion method, numerous novel soliton solutions are derived. These solutions, comprising hyperbolic, trigonometric and exponential functions, represent significant additions to the field of optics. Furthermore, we elucidate the physical characteristics of these novel optical soliton solutions by presenting a series of three-dimensional (3D) and two-dimensional (2D) graphs with appropriate parameter values. [ABSTRACT FROM AUTHOR]

The impact of deformation space on the physical characteristics of diverse physics systems has been thoroughly investigated in research papers. In this work, we study the deformed Klein–Gordon equation (DKGE) in the three-dimensional relativistic non-commutative quantum space (3D-RNCQS) regime by using the improved Hua plus modified Eckart potential (IHPMEP) model. For this consideration, the DKGE in the 3D-RNCQS regime is solved using the standard perturbation theory and the well-known Bopp’s shifts method with the Greene–Aldrich approximation to the centrifugal barrier. The new relativistic energy equation and eigenfunction for the IHPMEP in the presence of deformation space-space for the heterogeneous (CO, HF, and NO) and homogeneous (N2, H2, and Li2) diatomic molecules are obtained to be sensitive to the atomic quantum numbers (j,l,s, and m), the mixed potential depths (V0,V1,V2, and V3), the inverse of the screening parameter α, and non-commutativity parameters (Φ, χ, and ζ). Analysis is performed on the non-relativistic limit of new energy spectra. By appropriately adjusting the combined potential parameters, we analyze the obtained new bound state eigenvalues of the DKGE and deformed Schrödinger equation with the IHPMEP in 3D-NCQS symmetries and obtain the new modified Eckart potential, the modified Hua potential, the modified Morse potential, and the modified Pöschl–Teller potential. Within the framework of the 3D-NRNCQS regime, the homogeneous and heterogeneous composite systems under IHPMEP models are examined. A thorough investigation is carried out into the impact of space-space deformation on the thermal parameters of the IHPMEP, including the partition function, mean energy, free energy, specific heat, and entropy. This work is of a fundamental absorbability nature and pedagogical interest in atomic and molecular physics. [ABSTRACT FROM AUTHOR]

In this paper, we investigate a (3+1)-dimensional integrable fourth-order nonlinear equation that can model both the right- and left-going waves in a fluid. Based on the Hirota method, we derive three sets of the bilinear auto-Bäcklund transformations along with some analytic solutions. Through the extended homoclinic test approach, we obtain some breather solutions. We find that the breather propagates steadily along a straight line, with one hole and one peak in each period. Graphical investigation indicates that the coefficients in that equation affect the location and shape of the breather. Moreover, we construct some mixed lump–kink solutions. Fusion and fission between a lump wave and a kink soliton are analyzed graphically. The solutions addressed in this paper may be applied to mimic some complex waves in fluids. [ABSTRACT FROM AUTHOR]

In this paper, the nonlinear Klein–Gordon equation's exact solutions are obtained through the application of an appropriate transformation based on He's semi-inverse approach. This equation considered a generalization of other famous models in applied science, such as Phi-4 equation, Duffing equation, Fisher–Kolmogorov model through population dynamics and Hodgkin–Huxley equation that characterizes the propagation of electrical signals via nervous system. The suggested approach is simple, robust, and efficient, and its application in other partial differential equations in applied science seems promising. Numerical solution of the nonlinear Klein–Gordon equation is presented using finite difference method. The method's accuracy is demonstrated by contrasting it with the exact solution that we obtained. The Von Neumann stability technique is applied to obtain the stability condition and a convergent test is presented Some 2D and 3D graphs matching to chosen solutions are simulated by taking into account appropriate values for the parameters. [ABSTRACT FROM AUTHOR]

In this paper, we propose and analyze two highly efficient compact finite difference schemes for coupled nonlinear wave equations containing coupled sine-Gordon equations and coupled Klein-Gordon equations. To construct energy-preserving, high-order accurate and linear numerical methods, we first utilize the scalar auxiliary variable (SAV) approach and introduce three auxiliary functions to rewrite the original problem as a new equivalent system. Then we make use of the compact finite difference method and the Crank-Nicolson method to propose an efficient fully-discrete scheme (SAV-CFD-CN). The modified energy conservation and the convergence of the SAV-CFD-CN scheme are proved in detail, which has fourth-order convergence in space and second-order convergence in time. In order to preserve the discrete energy of original system, we further combine Lagrange multiplier approach, compact finite difference method and the Crank-Nicolson method to propose the second fully-discrete scheme (LM-CFD-CN). The proposed two schemes are high-order accurate, linear and highly efficient, only four symmetric positive definite systems with constant coefficients are required to be solved at each time level. Numerical experiments for the coupled nonlinear wave equations are given to confirm theoretical findings. [ABSTRACT FROM AUTHOR]

We show that only pseudo-spherical surface whose caustic becomes a minimal surface is Dini surface family. Moreover, we give the Weierstrass data for corresponding minimal surface to the caustic. [ABSTRACT FROM AUTHOR]

The nonlinear Schrödinger–Poisson equation with gravitational potential field is examined, along with its many soliton interactions and travelling wave solutions. By applying Hirota Bilinear scheme, we will give a brief study on transformation of solitons, also by Sine–Gordon expansion scheme solitary travelling wave solutions will be obtained in terms of trigonometric and hyperbolic functions. The innovative Hirota bilinear method is a powerful and standard technique, play a crucial role in producing soliton solutions as well as lump solutions. With the assistance of this method one can transformed integrable equation into Hirota bilinear form under dependent variable transformation. Soliton solutions obtained by Hirota bilinear method are significant in mathematical physics, may be superimposed in fibers. These solitons are applicable in optical communications, which enable to produce faster, richer, more secure and more flexible communication systems. Also we will investigate the gravitational potential's behaviour on soliton solutions, such as parabolic, bright, dark, anti-dark, combination and S-shaped solitons. Contour plots and three-dimensional soliton solutions are provided. These solitons are important in nonlinear processes, magnetic devices, and lasers beams because of variations in gravitational field. [ABSTRACT FROM AUTHOR]

In this work, we investigate the (2+1)-dimensional nonlinear electrical transmission line model (NETLM) and propose a modified generalized Riccati equation mapping (mGREM) approach for obtaining analytical soliton solutions. The NETLM characterizes wave distributions on network lines. We introduce the mGREM technique, which involves a traveling wave transformation to convert the equation into nonlinear ordinary differential equations. By applying this method, we extract various soliton solutions, including traveling waves, kink-type solitons, bell-shape solitons, and multi-solitons. These findings contribute to understanding the (2+1)-dimensional NETLM and provide valuable insights into wave distributions on network lines. To gain a comprehensive understanding of the solutions' dynamics, we conduct numerical simulations and present two-dimensional, three-dimensional, and contour graphs. These graphical illustrations offer valuable insights into voltage propagation patterns within the system. Our results contribute to the existing literature and open avenues for further investigations in the field of NETLM. The significance of this work lies in the successful application of the mGREM method to obtain diverse traveling wave solutions for the (2+1)-dimensional NETLM. By employing symbolic computation in Mathematica, we extract analytical soliton solutions, which go beyond previous efforts in the literature. The proposed approach provides a novel perspective on the NETLM equation and its soliton solutions. The obtained results enhance our understanding of the system's behavior and pave the way for future research in this domain. [ABSTRACT FROM AUTHOR]

The Klein–Gordon equation is a fundamental theoretical physics concept, governing the behavior of relativistic quantum particles with spin-zero. Its numerical solution is crucial in fields like quantum field theory, particle physics, and cosmology. The study explores numerical methodologies for solving this equation, highlighting their significance and challenges. This study uses the collocation method to approximate fractional Klein–Gordon models of distributed order based on Shifted Jacobi orthogonal polynomials and Shifted fractional order Jacobi orthogonal functions. While, the distributed term (integral term) was treat using Legendre–Gauss–Lobatto quadrature. It assesses residuals through finite expansion and yields accurate numerical results. The method is more factual and fair when initial and boundary conditions are enforced. Numerical simulations are presented to demonstrate the method's accuracy, particularly in fractional Klein–Gordon models of distributed order. Furthermore, we offer a few numerical test scenarios to show that the method is able to maintain the non-smooth solution of the underlying issue. [ABSTRACT FROM AUTHOR]

The Phi-Four (also embodied as ∅4) equation (PFE) is one of the most significant models in nonlinear physics, that emerges in particle physics, condensed matter physics and cosmic theory. In this study, propagating soliton solutions for the PFE were obtained by employing the extended direct algebraic method (EDAM). This transformational method reformulated the model into an assortment of nonlinear algebraic equations using a series-form solution. These equations were then solved with the aid of Maple software, producing a large number of soliton solutions. New families of soliton solutions, including exponential, rational, hyperbolic, and trigonometric functions, are included in these solutions. Using 3D, 2D, and contour graphs, the shape, amplitude, and propagation behaviour of some solitons were visualized which revealed the existence of kink, shock, bright-dark, hump, lump-type, dromion, and periodic solitons in the context of PFE. The study was groundbreaking as it extended the suggested strategy to the PFE that was being aimed at, yielding a significant amount of soliton wave solutions while providing new insights into the behavioral characteristics of soliton. This approach surpassed previous approaches by offering a systematic approach to solving nonlinear problems in analogous challenging situations. Furthermore, the results also showed that the suggested method worked well for building families of propagating soliton solutions for intricate models such as the PFE. [ABSTRACT FROM AUTHOR]

In this paper, we analyze the results for the ground state of a Klein–Gordon oscillator under gravitational effects in a topologically charged Ellis–Bronnikov wormhole derived recently. We show that the authors failed to truncate a power-series and, consequently, could not obtain the desired solution to a differential equation. For this reason, their analytical expression for the ground-state eigenvalue is incorrect, a fact that affects all the conclusions drawn by the authors for the physical problem. [ABSTRACT FROM AUTHOR]

This work is based upon classical achievements of S-matrix bootstrap together with an outline of its modern applications. We establish the connection from integral scattering amplitudes to the use of bootstrap technique in modern conformal field theories and holography. Using bootstrap approach, we provide the calculations for integral representations for the sine-Gordon S-matrix elements. Minimal form factors of this model are related to recent studies of certain conformal field theories. We also propose the extension of the sine-Gordon bootstrap equations to the space of conformal correlators known as celestial amplitudes. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

The goal of this research is to utilize some ansatz forms of solutions to obtain novel forms of soliton solutions for the Benney–Luke equation. It is a mathematically valid approximation that describes the propagation of two-way water waves in the presence of surface tension. By using ansatz forms of solutions, with an appropriate set of parameters, the lump soliton, periodic cross-kink waves, multi-waves, breather waves, Ma-breather, Kuznetsov–Ma-breather, periodic waves and rogue waves solutions can be obtained. Breather waves are confined, periodic, nonlinear wave solutions that preserve their amplitude and shape despite alternating between compression and expansion. For some integrable nonlinear partial differential equations, a lump soliton is a confined, stable solitary wave solution. Rogue waves are unusually powerful and sharp ocean surface waves that deviate significantly from the surrounding wave pattern. They pose a threat to maritime safety. They typically show up in solitary, seemingly random circumstances. Periodic cross-kink waves are a particular type of wave pattern that has frequent bends or oscillations that cross at right angles. These waves provide insights into complicated wave dynamics and arise spontaneously in a variety of settings. In order to predict the wave dynamics, certain 2D, 3D and contour profiles are also analyzed. Since these recently discovered solutions contain certain arbitrary constants, they can be used to describe the variation in the qualitative characteristics of wave phenomena. [ABSTRACT FROM AUTHOR]

Purpose: The purpose of the article is to conduct a mathematical and theoretical analysis of soliton solutions for a specific nonlinear evolution equation known as the (2 + 1)-dimensional Zoomeron equation. Solitons are solitary wave solutions that maintain their shape and propagate without changing form in certain nonlinear wave equations. The Zoomeron equation appears to be a special model in this context and is associated with other types of solitons, such as Boomeron and Trappon solitons. In this work, the authors employ two mathematical methods, the modified F-expansion approach with the Riccati equation and the modified generalized Kudryashov's methods, to derive various types of soliton solutions. These solutions include kink solitons, dark solitons, bright solitons, singular solitons, periodic singular solitons and rational solitons. The authors also present these solutions in different dimensions, including two-dimensional, three-dimensional and contour graphics, which can help visualize and understand the behavior of these solitons in the context of the Zoomeron equation. The primary goal of this article is to contribute to the understanding of soliton solutions in the context of the (2 + 1)-dimensional Zoomeron equation, and it serves as a mathematical and theoretical exploration of the properties and characteristics of these solitons in this specific nonlinear wave equation. Design/methodology/approach: The article's methodology involves applying specialized mathematical techniques to analyze and derive soliton solutions for the (2 + 1)-dimensional Zoomeron equation and then presenting these solutions graphically. The overall goal is to contribute to the understanding of soliton behavior in this specific nonlinear equation and potentially uncover new insights or applications of these soliton solutions. Findings: As for the findings of the article, they can be summarized as follows: The article provides a systematic exploration of the (2 + 1)-dimensional Zoomeron equation and its soliton solutions, which include different types of solitons. The key findings of the article are likely to include the derivation of exact mathematical expressions that describe these solitons and the successful visualization of these solutions. These findings contribute to a better understanding of solitons in this specific nonlinear wave equation, potentially shedding light on their behavior and applications within the context of the Zoomeron equation. Originality/value: The originality of this article is rooted in its exploration of soliton solutions within the (2 + 1)-dimensional Zoomeron equation, its application of specialized mathematical methods and its successful presentation of various soliton types through graphical representations. This research adds to the understanding of solitons in this specific nonlinear equation and potentially offers new insights and applications in this field. [ABSTRACT FROM AUTHOR]

In this study, we studied the (2+1)-dimensional complex modified Korteweg-de Vries (cmKdV) system using the improved Ricatti equation method. cmKdV are nonlinear and coupled partial differential equations that arise in various fields of applied science and engineering, such as ferromagnetic materials and optical fibers. When the method is applied to cmKdV, we successfully derive exact soliton solutions that accurately describe the wave propagation behavior of the system under consideration. The obtained results include trigonometric and hyperbolic function solutions. The results obtained are concise and offer a deeper insight into the dynamics and characteristics of cmKdV. Traveling wave solitons are plotted in 2D and 3D to demonstrate the wave propagation phenomena in the cmKdV model, which are in the form of kink, bright, dark, singular solitons, and periodic solitary wave structures. The method recovers many solutions compared with the existing methods in the literature, indicating that the proposed method is a powerful and valuable approach for achieving analytical solutions to a wide range of nonlinear partial differential equations [ABSTRACT FROM AUTHOR]

The Gilson–Pickering equation, which describes wave propagation in plasma physics and the structure of crystal lattice theory that is most frequently used. The discussed model is converted into a bilinear form utilizing the Hirota bilinear technique. Sets of case study are kink wave solutions; breather solutions; collision between soliton and periodic waves; soliton and periodic waves. The exp (- θ (α)) -function approach is employed to discover travelling wave solutions including five classes of solutions. In addition, it has been confirmed that the established findings are stable, and it has been helpful to validate the computations. The effect of the free variables on the behavior of obtained solutions to some plotted graphs for the exact cases is also explored depending upon the nature of nonlinearities. The exact solutions are utilized to demonstrate the physical natures of 3D, density, and 2D graphs. [ABSTRACT FROM AUTHOR]

This study delves into the analytical exploration of two pivotal equations, the confirmable Westervelt equation, relevant in acoustic nonlinear phenomena for applications like medical imaging and therapy, and the (2 + 1)-dimensional Wazwaz Kaur Boussinesq equation, providing insights into the unique characteristics of solitons and enriching our understanding of wave dynamics across various optical systems. Utilizing the potent ( G ′ / G , 1 / G )-expansion analytical method, we construct diverse wave structures and unveil a spectrum of soliton solutions, ranging from trigonometric and hyperbolic functions to rational expressions. Extensive validation using Mathematica software guarantees precision, while dynamic visual representations vividly portray a spectrum of soliton solutions. These solutions encompass a variety of patterns, such as bright solitons, kink solitons with periodic patterns, bell-shaped structures, parabolic structures, and hyperbolic formations. These solutions hold importance in acoustic image processing and the study of wave dynamics across different optical systems. They aid in comprehending the propagation of light in optical systems, thereby providing valuable insights that drive advancements in optical technology and communication. We also investigate modulation instability of the Wazwaz Kaur Boussinesq equation and stability analysis of the confirmable Westervelt equation. Our mentioned expansion scheme proves versatile and applicable across a diverse array of mathematical and physical challenges, showcasing its utility in producing such solutions. [ABSTRACT FROM AUTHOR]

The fundamental intention of this article is to find several soliton solutions to the generalized Benjamin–Bona–Mahony equation. The fractional derivatives are described in the conformable sense. These solutions are generated using the functional variable approach. Then, we acquire via this technique some distinctive solutions, such as the singular soliton, periodic soliton, bell-shaped soliton, and kink soliton solutions. The fundamental feature of the demonstrated method is that it's convenient and provides precise, effective solutions to nonlinear equations. The solutions given will offer a thorough examination of this model and any associated occurrences. [ABSTRACT FROM AUTHOR]

Nonlinear partial integro-differential equations (PIDEs) are applied to present the various practical phenomena in a multitude of sectors of modern science and engineering, especially in optic fiber, quantum electronics, modern physics, and the special field of nonlinear wave motion. Basically, our research demonstrates a way to generate a significant quantity of solutions to these types of two PIDEs. In this research, we have used an efficient mathematical tool namely the generalized G ′ / G -expansion method to acquire the closed form soliton solutions for the (2 + 1)-dimensional first integro-differential Kadomtsev–Petviashivili (KP) hierarchy equation and the (2 + 1)-dimensional second integro-differential KP hierarchy equation utilizing a code likely Mathematica. The explicit closed form soliton solutions of these two PIDEs are found in the pattern of trigonometric, hyperbolic, and rational functions which are compared to all the well-known results that are yielded in the paper. We attain solutions that are graphically displayed in addition to physically described in 3D structure, contour, and 2D, such as a bell-shaped soliton, a singular bell-shaped soliton, and some kink-shaped solitons. As far as the authors' wisdom, the outcomes of these problems gained by the offered expansion method are renewed closed form solitary wave and investigated here for the first time. The analysis of obtained results will be able to provide a constructive explanation of the physical phenomena in optical physics and engineering. [ABSTRACT FROM AUTHOR]

In this paper, the new solitary wave solutions of the (3+1)-dimensional Boiti-Leon-Manna-Pempinelli equation are obtained by Lie group symmetry method and the extended homoclinic test approach. Firstly, the equation can be reduced to (1+1)-dimensional partial differential equation by Lie group symmetry, and corresponding bilinear forms of the equation are given by symmetry functions. Secondly, the extended homoclinic test approach is employed to obtain the new kink type and singular solitary wave solutions. In addition, some new traveling and non-traveling wave solutions with arbitrary functions and oscillating tail are investigated through the special transformations for the (3+1)-dimensional Boiti-Leon-Manna-Pempinelli equation. • The two different bilinear forms are obtained. • Some new solitary wave solutions are given. • Some new non-traveling wave solutions and their different behaviors are investigated. [ABSTRACT FROM AUTHOR]

This paper presents Hamiltonian finite element methods for approximating semilinear wave propagation problems, including the nonlinear Klein–Gordon and sine-Gordon equations. The aim is to obtain accurate high-order approximations while conserving physical quantities of interest such as energy. To achieve conservation properties at a discrete level, we propose semidiscrete schemes based on two Hamiltonian structures of the equation. These include Mixed finite element methods, discontinuous Galerkin methods, and hybridizable discontinuous Galerkin methods (HDG). In particular, we propose a new class of DG methods using time operators to define the numerical traces, ultimately leading to an energy-conserving scheme. Time discretization uses Symplectic explicit-partitioned and diagonally-implicit Runge–Kutta schemes. Furthermore, the paper showcases several numerical examples that demonstrate the accuracy and energy conservation properties of the approximations, along with the simulation of soliton cloning. [ABSTRACT FROM AUTHOR]

In this paper, the one-dimensional nonlinear Sine–Gordon equation is solved using the "Exponential modified cubic B-spline differential quadrature method with the leave-one-out cross-validation (LOOCV) approach". By employing the LOOCV approach to determine the optimal value of the parameter ϵ involved in the basis function, the accuracy and effectiveness of the results are improved. The combination of this approach with the exponential modified cubic B-spline differential quadrature method, which is novel in the literature, is likely to attract researchers' interest. Additionally, the procedure is implemented on six examples of the Sine–Gordon equation. The results are presented in the form of tables and figures. It is demonstrated that this approach is straightforward and yields superior outcomes compared to the existing literature. This paper also presents an insightful discussion on the significant application of the Sine–Gordon equation in Josephson junctions and its crucial role in new technologies. [ABSTRACT FROM AUTHOR]

Abstract To examine the dynamical behavior of travelling wave solutions of the water wave phenomenon for the family of 3D fractional Wazwaz-Benjamin-Bona-Mahony (WBBM) equations, this work employs the rational Sine-Gordon expansion (RSGE) approach based on the conformable fractional derivative. The method generalizes the well-known sine-Gordon expansion using the sine-Gordon equation as an auxiliary equation. In contrast to the conventional sine-Gordon expansion method, it takes a more general approach, a rational function rather than a polynomial one of the solutions of the auxiliary equation. The method described above is used to generate various solutions of the WBBM equations for hyperbolic functions, including soliton, singular soliton, multiple-soliton, kink, cusp, lump-kink, kink double-soliton, etc. The RSGE method contributes to our understanding of nonlinear phenomena, provides exact solutions to nonlinear equations, aids in studying solitons, advances mathematical techniques, and finds applications in various scientific and engineering disciplines. The answers are graphically shown in three-dimensional (3D) surface plots and contour plots using the MATLAB program. The resolutions of the equation, which have appropriate parameters, exhibit the absolute wave configurations in all screens. Furthermore, it can be inferred that the physical characteristics of the discovered solutions and their features may aid in our understanding of the propagation of shallow water waves in nonlinear dynamics.

In this paper, the initial boundary value problem for Klein–Gordon equation with weak and strong damping terms and nonlinear logarithmic term is investigated, which is known as one of the nonlinear wave equations in relativistic quantum mechanics and quantum field theory. Firstly, we prove the local existence and uniqueness of weak solution by using the Galerkin method and Contraction mapping principle. The global existence, energy decay and finite time blow-up of the solution with subcritical initial energy are established. Then these conclusions are extended to the critical initial energy. Besides, the finite time blow-up result with supercritical initial energy is shown. [ABSTRACT FROM AUTHOR]

In this paper, a compound equation which is a mix of the Wadati--Konno--Ichikawa (WKI) equation and the short-pulse (SP) equation is first studied. By transforming both the independent and dependent variables in the equation, we introduce a novel hodograph transformation to convert the compoundWKI--SP equation into the mKdV--SG (modified Korteweg--de Vries and sine- Gordon) equation. The multiloop soliton solutions in the form of the parametric representation are found. It is shown that the N-loop soliton solution may be decomposed exactly into N separate soliton elements by using a Moloney--Hodnett-type decomposition. By virtue of the decomposed soliton solutions, the asymptotic behaviors of N = 2 and N = 3 are investigated in detail. The corresponding phase shifts of each loop or antiloop soliton caused by its interaction with the other ones are calculated. Furthermore, a new hierarchy of WKI--SP-type equations possessing multiloop soliton solutions is constructed. These deduced equations are all with time-varying coefficients and the corresponding dispersion relation will have a time-dependent velocity. The whole hierarchy of equations which include the WKI-type equations, the SP-type equations, and the compound generalized WKI--SP equations, are illustrated Lax integrable. The specific equation in the hierarchy is labeled as WKI--SP(n,m) equation so that its Lax pairs can be directly written out with the help of n and m. A unified hodograph transformation is established to relate the compound WKI--SP hierarchy with the mKdV--SG hierarchy. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the backward asymptotic autonomy of pullback attractors for asymptotically autonomous processes. Namely, time-components of the pullback attractors semi-converge to the global attractors of the corresponding limiting semigroups as the time-parameter goes to negative infinity. The present article is divided into two parts: theories and applications. In the theoretical part, we establish a sufficient and necessary criterion with respect to the backward asymptotic autonomy via backward compactness of pullback attractors. Moreover, this backward asymptotic autonomy is considered by the periodicity of pullback attractors. As for the applications part, we apply the abstract results to non-autonomous retarded sine-Gordon lattice systems. By backward uniform tail-estimates of solutions, we prove the existence of a pullback and global attractor for such lattice systems such that the backward asymptotic autonomy is satisfied. Furthermore, it is also fulfilled under the assumptions of the periodicity for the non-delay forcing and the convergence for processes. [ABSTRACT FROM AUTHOR]

The examination of new (3 + 1)-dimensional wave equations is undertaken in this study. Initially, the identification of the Lie symmetries of the model is carried out through the utilization of the Lie symmetry approach. The commutator and adjoint table of the symmetries are presented. Subsequently, the model under discussion is transformed into an ordinary differential equation using these symmetries. The construction of several bright, kink, and dark solitons for the suggested equation is then achieved through the utilization of the new auxiliary equation method. Subsequently, an analysis of the dynamical nature of the model is conducted, encompassing various angles such as bifurcation, chaos, and sensitivity. Bifurcation occurs at critical points within a dynamical system, accompanied by the application of an outward force, which unveils the emergence of chaotic phenomena. Two-dimensional plots, time plots, multistability, and Lyapunov exponents are presented to illustrate these chaotic behaviors. Furthermore, the sensitivity of the investigated model is executed utilizing the Runge–Kutta method. This analysis confirms that the stability of the solution is minimally affected by small changes in initial conditions. The attained outcomes show the effectiveness of the presented methods in evaluating solitons of multiple nonlinear models. [ABSTRACT FROM AUTHOR]

The purpose of this study is to address the two-dimensional counter-current capillary dominant imbibition of a wetting phase into a water-wet porous cylindrical matrix block partially submerged in the wetting phase. A two-dimensional unsteady-state diffusion equation is used to model the process. The governing equation is solved using a combination of the Laplace and the finite Fourier sine transforms to find and analyze the solutions for the normalized water saturation and the volume of the imbibed wetting phase. The results reveal that the volume of the imbibed wetting phase and the capillary diffusion shape factor for a partially submerged matrix block are significantly lower compared to those of a fully submerged matrix block, highlighting the overestimation of imbibed volume using available models based on full immersion in the wetting phase. It has been observed that the volume of the imbibed wetting phase increases over time until reaching a state of equilibrium. In the case of a partially submerged matrix block, the shape factor is inversely proportional to the square root of time (σ ∼ 1 / t ) during the early time and decreases sharply as the imbibed wetting phase reaches an equilibrium. In the case of a fully submerged matrix block, the shape factor is inversely proportional to the square root of time (σ ∼ 1 / t ) during the early time and later reaches a pseudo-steady-state value. The proposed model, along with the findings obtained, advances our understanding of capillary imbibition in porous media. [ABSTRACT FROM AUTHOR]

This research extensively investigates the fractional coupled Konno–Oono model, a prevalent framework in diverse scientific and engineering disciplines. The primary objective is to unravel the intricate dynamics embedded in this model. We adeptly convert partial differential equations into ordinary differential equation by applying appropriate transformations. Through the innovative approaches of the modified Khater method and the modified extended auxiliary equation method, we systematically derive solutions, encompassing periodic, dark, bright, singular, kink, and anti-kink solitons. Notably, these outcomes hold significant applications in the realm of magnetic fields. To assess the system's stability, we employ the linear stability mechanism, offering insights into modulation instability gain. Careful selection of parametric values allows us to conduct numerical simulations, presenting results through visually engaging figures that facilitate a deeper understanding of their significance. Importantly, our findings affirm that the applied techniques not only yield complete and uniform responses but also showcase simplicity, effectiveness, and remarkable computational efficiency. It is crucial to highlight that all the results presented in this study are obtained using the Mathematica software, emphasizing the reliability and precision of our computational approach. Overall, our research contributes valuable insights into the behavior of the fractional coupled Konno–Oono model, paving the way for enhanced comprehension and potential applications in magnetic field studies. [ABSTRACT FROM AUTHOR]

The (3+1)-dimensional double sine-Gordon equation plays a crucial role in various physical phenomena, including nonlinear wave propagation, field theory, and condensed matter physics. However, obtaining exact solutions to this equation faces significant challenges. In this article, we successfully employ a modified G ′ G 2 -expansion and improved tan ϕ ξ 2 -expansion methods to construct new analytical solutions to the double sine-Gordon equation. These solutions can be divided into four categories like trigonometric function solutions, hyperbolic function solutions, exponential solutions, and rational solutions. Our key findings include a rich spectrum of soliton solutions, encompassing bright, dark, singular, periodic, and mixed types, showcasing the (3+1)-dimensional double sine-Gordon equation ability to model diverse wave behaviors. We uncover previously unreported complex wave structures, revealing the potential for complex nonlinear interactions within the (3+1)-dimensional double sine-Gordon equation framework. We demonstrate the modified G ′ G 2 -expansion and improved tan ϕ ξ 2 -expansion methods effectiveness in handling higher-dimensional nonlinear partial differential equations, expanding their applicability in mathematical physics. These method offers enhanced flexibility and broader solution categories compared to conventional approaches. [ABSTRACT FROM AUTHOR]