sinh-Gordon Equation Research Articles

Stability analysis, modulation instability, and the analytical wave solitons to the fractional Boussinesq-Burgers system

Abstract The research is concerned with the novel analytical solitons to the (1+1)-D nonlinear Boussinesq-Burgers System (B-B S) in the sense of a new definition of fractional derivatives. The concerned system is helpful to describes the waves in different phenomenon,
including proliferation of waves in shallow water, oceanic waves and many others. Authors gain the solutions involving trigonometric, hyperbolic, and rational functions by using the expa function and the extended sinh-Gordon equation expansion (EShGEE) methods. Fractional derivative provides the better results than the present results. These results are helpful and useful in the different areas of applied sciences. The solutions are shown by 2-dimensional, 3-dimensional, and contour graphs. The solutions are useful in further studies of the governing model. The stability process is performed to verify that the solutions are exact and accurate. The modulation instability is used to determine the steady-state stable results to the governing equation. The techniques utilized are both simple and effective.

Harmonic maps between pseudo-Riemannian surfaces

We study locally harmonic maps between a Riemann surface or a Lorentz surface M and a Riemann or Lorentz surface N. All four cases are written using a unified formalism. Therefore properties and solutions to the harmonic map problem can be studied in a unified way.It is known that harmonic maps between Riemannian surfaces are classified by the classification of the solutions of a sinh-Gordon equation. We extend this result to all the four cases of harmonic maps between Riemannian or pseudo-Riemannian surfaces. The calculation of the corresponding harmonic map can be calculated by the solutions of the corresponding Beltrami equations in all the cases.We study the one-soliton solutions of this equation and we find the corresponding harmonic maps in a unified way.Next, we discuss a Bäcklund transformation of the harmonic map equations that provides a connection between the solutions of two sine or sinh-Gordon type equations. Finally, we give an example of a harmonic map that is constructed by the use of a Bäcklund transformation.

The Discovery of Truncated M-Fractional Exact Solitons and a Qualitative Analysis of the Generalized Bretherton Model

This paper is concerned with the novel exact solitons for the truncated M-fractional (1+1)-dimensional nonlinear generalized Bretherton model with arbitrary constants. This model is used to explain the resonant nonlinear interaction between the waves in different phenomena, including fluid dynamics, plasma physics, ocean waves, and many others. A series of exact solitons, including bright, dark, periodic, singular, singular–bright, singular–dark, and other solitons are obtained by applying the extended sinh-Gordon equation expansion (EShGEE) and the modified (G′/G2)-expansion techniques. A novel definition of fractional derivative provides solutions that are distinct from previous solutions. Mathematica software was used to obtain and verify the solutions. The solutions are shown through 2D, 3D, and density plots. A stability process was conducted to verify that the solutions are exact and accurate. Modulation instability was used to determine the steady-state results for the corresponding equation.

A new shifted generalized Chebyshev approach for multi-dimensional sinh-Gordon equation

The numerical treatment of multi-dimensional non-linear sinh-Gordon equations is the focus of this paper. We numerically solve the (1 + 1) and (2 + 1) sinh-Gordon equations using two collocation algorithms. We select the set of basis functions as a set of generalized Chebyshev polynomials (CPs), which we express as orthogonal combinations of CPs. We develop and utilize some formulas related to these polynomials to propose our numerical algorithms. Specific values for the high-order derivatives of the basis functions serve in the derivation of the two presented algorithms. Additionally, we provide an estimation of the basis functions used in the convergence analysis study. We follow the two collocation algorithms to transform the sinh-Gordon equations into non-linear equation systems, which any suitable solver can handle. We provide some examples and comparisons to confirm the effectiveness of our presented algorithms.

Investigation of the complex dynamical structure of bifurcation and dark soliton solutions to fractional generalized double [formula omitted]-Gordon equation

In the current research work, the classical wave equation is combined with a nonlinear sinh source term in the sinh-Gordon equation. It has been used in several scientific domains such as differential geometry theory, integrable quantum field theory, kink dynamics, and statistical mechanics. It makes more comprehensible the dynamics of strings and multi-strings in the constant curvature space. The current study has three main objectives. Examine the governing model to get novel solutions by employing the modified Kudryashov technique. Then compare it with the numerical technique of modified variational iteration method (MVIM) to calculate the error terms. Furthermore, employing bifurcation theory to produce a dynamical system. Additionally, use the dynamical system’s sensitivity analysis to investigate the model’s sensitivity. At last, for the validation of acquired results, the cryptography technique of novel image encryption and decryption is used. The research is greatly enhanced by the presentation of thorough 2D and 3D phase portraits. The field of mathematics and other sciences will benefit from these discoveries.

Solitary waves of M-fractional low-pass nonlinear electrical transmission line model arising in network system

In this study, we analyze the solitary wave behavior of a truncated M-fractional low-pass nonlinear electrical transmission line (NLETLs) model. NLETL models are relevant to computer network systems, particularly for high-speed data transmissions. They influence the behavior of signals traveling through network cables. To investigate the dynamics of solitary waves in the model, we applied the modified Sardar sub-equation and extended the sinh-Gordon equation expansion methods. We illustrated the 2D, 3D, and contour shapes of selected solutions for appropriate values of the NLETLs dynamics using Mathematica-14. Kink, anti-kink, bright-dark bell, dark bell, M-shaped periodic soliton, and logarithmic wave solutions were obtained. The results indicate that the proposed techniques may provide valuable, powerful, and efficient insights into the dynamics of nonlinear evolution models. The role of the fractional order derivative in making optical solutions is investigated in detail, which opens up opportunities for the creation of more complex models that can more accurately simulate optical phenomena in the real world.

Multiple soliton and traveling wave solutions for the negative-order-KdV-CBS model

The (3+1)-dimensional new negative-order-KdV-CBS model is investigated in this study. The suggested model combines the Korteweg-de Vries (KdV) and Calogero-Bogoyavlenskii-Schiff (CBS) equations. This research provides multiple soliton solutions and traveling wave solutions for the KdV-CBS model. Multiple exp-function methods have been used for extracting soliton solutions. For this aim, the extended sinh-Gordon equation expansion approach was selected to get traveling wave solutions. The findings are graphically examined by selecting appropriate values for arbitrary parameters.

Global Cauchy problem for the complex NLKG and sinh-Gordon equations in super-critical spaces

By introducing a class of new function spaces Bp,qσ,s as the resolution spaces, we study the Cauchy problem for the nonlinear Klein-Gordon (NLKG) and sinh-Gordon equations in all spatial dimensions d⩾1,∂t2u+u−Δu+f(u)=0,(u,∂tu)|t=0=(u0,u1), where f(u)=u1+α or f(u)=sinh⁡u−u. We consider the initial data (u0,u1) in super-critical function spaces Eσ,s×Eσ−1,s for which their norms are defined by‖f‖Eσ,s=‖〈ξ〉σ2s|ξ|fˆ(ξ)‖L2,s<0,σ∈R. Any Sobolev space Hκ can be embedded into Eσ,s, i.e., Hκ⊂Eσ,s for any κ,σ∈R and s<0. We show the global existence and uniqueness of the solutions of NLKG if the initial data belong to some Eσ,s×Eσ−1,s (s<0,σ⩾max⁡(d/2−2/α,1/2),α∈N,α⩾4/d) and their Fourier transforms are supported in the first octant, the smallness conditions on the initial data in Eσ,s×Eσ−1,s are not required for the global solutions. Similar results hold for the sinh-Gordon equation if the spatial dimensions d⩾2.

Phase characterization and exraction of new forms of solitons for the (3+1)-dimensional q-deformed Sinh-Gordon equation

In this article, the (3+1)-dimensional q-deformed Sinh-Gordon model is investigated to extract analytical solutions using the unified method. This technique effectively extracts polynomial and rational function solutions. When the appropriate limiting constraints are given to the parameters, this technique successfully retrieves hyperbolic and trigonometric results. Some graphical representations of the solutions of the proposed equation are illustrated. Additionally, all feasible phase portraits are shown, the planer dynamical system of the equation under discussion is built using Galilean transformation, and sensitive inspection is used to verify the sensitivity of the equation under consideration. There aren't many previous methods for solving this kind of equation, either analytically or numerically. This work is highly valuable for the understanding of various symmetrical physical systems.

Unveiling novel dynamics in Q-deformed Sinh-Gordon model: exploring explicit wave solutions and stability analysis

The new generalized Q-deformed Sinh-Gordon equation is investigated in this article. Numerous physical interpretations and applications exist for the said model in several scientific fields. Therefore, for analyzing the underlying physical system, it is imperative to look at the traveling wave behavior displayed by this equation. To complete our thrust three innovative integration architectures including the modified sub-equation method, the G′/(bG′+G+a) -expansion method and G′G2 -expansion method have been utilized to create precise closed form traveling wave solutions. These methods provide solutions in the form of hyperbolic function solutions, trigonometric function solutions and rational solutions. The obtained solutions are discussed with 3D surface plots and 2D line plots to visualize and support the theoretical results. Furthermore, stability for the model is also discussed. This paper discusses a major development as well as gradual improvements to previous works on the current model. Moreover, we discuss the stability analysis of the studied equation and the outcomes are visually depicted in graphical form. To authenticate the solutions provided in this manuscript, each solution undergoes a meticulous validation process.

Explicit solitary wave profiles and stability analysis of biomembranes and nerves

This paper examines the stability analysis and exact solitary wave solutions of the nonlinear partial differential equation known as the Heimburg model. The several types of solitary wave solutions, soliton solutions and Jacobi elliptic doubly periodic function solutions are explored by using the extended Sinh-Gordon equation expansion approach. These investigations exhibit the system’s astounding diversity of waveforms, highlighting its potential applications in nerves and biomembranes. By selecting some appropriate values for the parameters, 3D, 2D, and its corresponding contour graph are plotted to represent the physical relevance of some of the solutions. Additionally, the linearized stability of this system is analyzed. The suggested approach is the finest resource for the analytical investigation of any nonlinear issue that occurs in various scientific fields.

Traveling wave solutions, numerical solutions, and stability analysis of the (2+1) conformal time-fractional generalized q-deformed sinh-Gordon equation

Abstract The two-dimensional conformal time-fractional generalized q q -deformed sinh-Gordon equation has been used to model a variety of physical systems, including soliton propagation in asymmetric media, nonlinear waves in optical fibers, quantum field theory, and condensed matter physics. The equation is able to capture the complex dynamics of these systems and has been shown to be a powerful tool for studying them. This article discusses the two-dimensional conformal time-fractional generalized q q -deformed sinh-Gordon equation both analytically and numerically using Kudryashov’s approach and the finite difference method. In addition, the stability analysis and local truncation error of the equation are discussed. A number of illustrations are also included to show the various solitons propagation patterns. The proposed equation has opened up new possibilities for modeling asymmetric physical systems.

Reducing Modified Formulas for Sinh-Gordon Equation to the Painleve Equations

Reducing Modified Formulas for Sinh-Gordon Equation to the Painleve Equations

Soliton solutions of the improved perturbed nonlinear Schrödinger equation having parabolic law with non-local nonlinearity in the presence of chromatic and spatio-temporal dispersion terms

This paper was organized to examine the analytical solutions of the improved perturbed parabolic-law nonlinear Schrodinger equation including non-local nonlinearity in the presence of chromatic and spatio-temporal dispersion. This model mostly makes use of studying the propagation of optical pulses in fiber optic communication systems. We performed the Sinh-Gordon equation expansion method so that we produce the analytical solutions of the model under consideration. It was confirmed that the acquired solutions satisfy the main model. Therefore, bright and dark soliton solutions were retrieved; besides, various 3D and 2D graphical illustrations of the solitons were demonstrated via appropriate values of the parameters. Furthermore, this manuscript focused on the parameters’ effect on the acquired solitons behavior.

Exploring exact solutions for physical differential models through generalized derivatives

In this study, we develop three well-known fractional differential physical models with novel exact solutions. Liouville, Dodd-Bullough-Mikhailov (DBM), and Sinh-Gordon equations are the models under consideration. These models will be broken down into three nonlinear ordinary differential equations using a waveform transformation, which can be precisely solved using the approach of the simplest equation method. The suggested method is applicable to several categories of nonlinear physical models and allows us to extract numerous generalized solutions in soliton and periodic forms The resulting solutions may also be directly compared with a number of findings obtained in the literature. Additionally, representations in two and three dimensions are provided to show how changing the fractional parameter’s amount may impact how monotonic the solutions are obtained.

Superposition Formulae for the Geometric Bäcklund Transformations of the Hyperbolic and Elliptic Sine-Gordon and Sinh-Gordon Equations

We provide superposition formulae for the six cases of Bäcklund transformations corresponding to space-like and time-like surfaces in the 3-dimensional pseudo-Euclidean space. In each case, the surfaces have constant negative or positive Gaussian curvature and they correspond to solutions of one of the following equations: the sine-Gordon, the sinh-Gordon, the elliptic sine-Gordon and the elliptic sinh-Gordon equation. The superposition formulae provide infinitely many solutions algebraically after the first integration of the Bäcklund transformation. Such transformations and the corresponding superposition formulae provide solutions of the same hyperbolic equation, while they show an unusual property for the elliptic equations. The Bäcklund transformation alternates solutions of the elliptic sinh-Gordon equation with those of the elliptic sine-Gordon equation and the superposition formulae provide solutions of the same elliptic equation. Explicit examples and illustrations are given.

Investigation of complex hyperbolic and periodic wave structures to a new form of the q-deformed sinh-Gordon equation with fractional temporal evolution

This paper presents the fractional generalized q-deformed sinh-Gordon equation. The fractional effects of the temporal derivative of the proposed model are studied using a conformable derivative. The analytical solutions of the governing model depend on the specified parameters. The resulting equation is studied with two integration architectures: the sine-Gordon expansion method and the modified auxiliary equation method. These strategies extract hyperbolic, trigonometric, and rational form solutions. For appropriate parametric values and different values of fractional parameter α, the acquired findings are displayed via 3D graphics, 2D line plots, and contour plots. The graphical simulations of the constricted solutions depict the existence of bright soliton, dark soliton, and periodic waves. The considered model is useful in describing physical mechanisms that possess broken symmetry and incorporate effects such as amplification or dissipation.

Self–dual solutions of a field theory model of two linked rings

In this work the connection established in [7,8] between a model of two linked polymers rings with fixed Gaussian linking number forming a 4-plat and the statistical mechanics of non-relativistic anyon particles is explored. The excluded volume interactions have been switched off and only the interactions of entropic origin arising from the topological constraints are considered. An interpretation from the polymer point of view of the field equations that minimize the energy of the model in the limit in which one of the spatial dimensions of the 4-plat becomes very large is provided. It is shown that the self-dual contributions are responsible for the long-range interactions that are necessary for preserving the global topological properties of the system during the thermal fluctuations. The non self-dual part is also related to the topological constraints, and takes into account the local interactions acting on the monomers in order to prevent the breaking of the polymer lines. It turns out that the energy landscape of the two linked rings is quite complex. Assuming as a rough approximation that the monomer densities of half of the 4-plat are constant, at least two points of energy minimum are found. Classes of non-trivial self-dual solutions of the self-dual field equations are derived. One of these classes is characterized by densities of monomers that are the squared modulus of holomorphic functions. The second class is obtained under some assumptions that allow to reduce the self-dual equations to an analog of the Gouy-Chapman equation for the charge distribution of ions in a double layer capacitor. In the present case, the spatial distribution of the electric potential of the ions is replaced by the spatial distribution of the fictitious magnetic fields associated with the presence of the topological constraints. In the limit in which two of the spatial dimensions are large in comparison with the third one, we provide exact formulas for the conformations of the monomer densities of the 4-plat by using the elliptic, hyperbolic and trigonometric solutions of the sinh-Gordon and cosh-Gordon equations which have been used for instance in the construction of classical string solutions in AdS3 and dS3 [9].

Novel closed-form analytical solutions and modulation instability spectrum induced by the Salerno equation describing nonlinear discrete electrical lattice via symbolic computation

A nonlinear electrical lattice is an effective experimental tool for dealing with nonlinear dispersive media and creating possibilities for the realistic modeling of electrical soliton. In this work, we discuss the dynamical behavior of the governing (1+1)-dimensional time–space Salerno equation, which describes the discrete electrical lattice with nonlinear dispersion. The different kinds of solutions in the form of periodic, Jacobi elliptic, and exponential functions are obtained by the two analytical approaches, namely the modified generalized exponential rational function method (MGERFM) and the extended sinh-Gordon equation expansion method (shGEEM). The MGERF approach is applied to construct exact solitary wave solutions containing trigonometry, hyperbolic and exponential functions, while the extended shGEE technique is employed to obtain periodic wave solutions containing hyperbolic and Jacobi elliptic functions via performing the symbolic computation in the software package MATHEMATICA. Moreover, by means of standard linear-stability analysis, the modulation instability (MI) and the MI gain spectrum is analytically computed for the considered equation. By selecting suitable parametric values, solution profiles are portrayed in order to make the solutions physically relevant. These results yield periodic waves, kink waves, multi-soliton, and their interactions. By comparing the obtained solutions to the previous findings, we can reveal the novelty of the solutions. In addition, all the obtained solutions were verified by substituting them back into the governing equation using the Mathematica software. The acquired results are significant for the study of wave propagation, signal transmission, and applications of super-transmission phenomena.

Chaotic behavior and construction of a variety of wave structures related to a new form of generalized q-Deformed sinh-Gordon model using couple of integration norms

&lt;abstract&gt;&lt;p&gt;The generalized q-deformed sinh Gordon equation (GDSGE) serves as a significant nonlinear partial differential equation with profound applications in physics. This study investigates the GDSGE's mathematical and physical properties, examining its solutions and clarifying the essence of the q-deformation parameter. The Sardar sub-equation method (SSEM) and sine-Gordon expansion method (SGEM) are employed to solve this GDSGE. The synergistic application of these techniques improves our knowledge of the GDSGE and provides a thorough foundation for investigating different evolution models arising in various branches of mathematics and physics. A positive aspect of the proposed methods is that they offer a wide variety of solitons, including bright, singular, dark, combination dark-singular, combined dark-bright, and periodic singular solitons. Obtained solutions demonstrate the method's high degree of reliability, simplicity, and functionalization for various nonlinear equations. To better describe the physical characterization of solutions, a few 2D and 3D visualizations are generated by taking precise values for parameters using mathematical software, Mathematica.&lt;/p&gt;&lt;/abstract&gt;