Types Of Soliton Solutions Research Articles

Analysis of new soliton type solutions to generalized extended (2 + 1)-dimensional Kadomtsev-Petviashvili equation via two techniques

In this paper, we examine the solitary wave solutions of the generalised (2 + 1) extended Kadomtsev–Petviashvili (eKP) equation, which is used as model for the surface waves and internal waves in straits or channels and describes the dynamics of nonlinear waves in plasma physics and fluid dynamics. We secure distinct soliton solutions by employing two amelioration methodologies, namely the improved F-expansion method and the new Kudryashov method. These methods explore several unique forms of solitary wave solutions, including dark, rational, singular, periodic, and exponential ones. In order to illustrate propagation of some reported wave solutions, the graphs associated with these solutions are presented by choosing appropriate parametric values in 2D, 3D, contour, and density plots with the use of the symbolic software Mathematica 13.0. The presented solutions realize very significant fact for investigation the qualitative interpretations for various phenomena in our nature. The computed solutions revealed that the applied methods are influential, efficious and skilful and can be a best way to handle other higher order nonlinear evolution equations.

Construction of fractional granular model and bright, dark, lump, breather types soliton solutions using Hirota bilinear method

The present article designs the granular metamaterials considering the granular structures of discrete particles which are different from elastic metamaterials consisting of continuous media. In granular metamaterials, the wave propagates through contact with neighboring particles. To identify the propagating properties of wave quantities in the rough granular medium the fractional granular equation is formulated directly in a pre-compressed spherical chain adopting Hertz law and long wave approximation theory. Using phase and group velocities, Caputo fractional derivatives are used to illustrate normal and anomalous dispersion wave dependence. To demonstrate in depth the dynamical behavior of the wave profile, various types of complex solutions like multi-shock, multi-solitons, lump, and breather solutions of the one-dimensional time fractional granular equations are explored employing Hirota’s bilinear approach. Finally, the more complicated hybrid solutions such as kink with the lump, soliton with the lump, etc. are exhibited from numerical understanding. The numerical graphs and figures demonstrate the crucial role of the order of derivative (roughness parameter) in the formation of different types of soliton solutions.

Solitons in openN= 2 string theory

AbstractThe open N = 2 string theory is defined on the 4D space-time with the split signature (+, +, −, −). The string field theory action of the open N = 2 string theory is described by the 4D Wess–Zumino–Witten (WZW4) model. The equation of motion of the WZW4 model is the Yang equation, which is equivalent to the anti-self-dual Yang–Mills equation. In this paper, we study soliton-type classical solutions of the WZW4 model in the split signature by calculating the action density of the WZW4 model. We find that the action density of the one-soliton solutions is localized on a 3D hyperplane. This shows that there would be codimension-one-solitonic objects, or equivalently some kind of three-branes in the open N = 2 string theory. We also prove that, in the asymptotic region of the space-time, the action density of the n-soliton solutions is a “non-linear superposition” of n one-solitons. This suggests the existence of n intersecting three-branes in the N = 2 strings. Finally we make a reduction to a (1 + 2)D real space-time to calculate the energy densities of the soliton solutions. We can successfully evaluate the energy distribution for the two-soliton solutions and find that there is no singularity in the interacting region. This implies the existence of smooth intersecting codimension-one branes in the whole region. Soliton solutions in the Euclidean signature are also discussed.

Dynamical behaviors with various exact solutions to a (2+1)-dimensional asymmetric Nizhnik–Novikov–Veselov equation using two efficient integral approaches

The main goal of this research work is to explore novel soliton solutions and dynamics of solitonic structures to asymmetric Nizhnik–Novikov–Veselov (ANNV) equations by two methods, namely, the generalized exponential rational function (GERF) method and the modified extended tanh expansion (METE) method. These techniques are the most effective and reliable tools for solving nonlinear equations with partial derivatives. The used methods provide a wide range of soliton solutions that can motivate applied scientists and researchers for these specific structures. These acquired solutions are in the form of exponential, trigonometric and hyperbolic functions including, [Formula: see text] [Formula: see text] and of their combinations. It has also been observed that the obtained soliton solutions of the ANNV equations are bell-shape, anti-bell-shape, periodic solution, multisoliton and different types of soliton solutions. To illustrate the physical features and dynamics characteristics of some obtained solutions, three-dimensional (3D) and two-dimensional (2D) figures are exhibited through the Mathematica software. The methods applied in this paper are suitable to study the soliton solutions for the ANNV equation without producing the complexities in some other known analytical method. Finally, the employed methods can be earmarked easily for solving various classes of nonlinear evolution equations appearing in mathematical physics, fluid dynamics, plasma physics, nonlinear waves and nonlinear sciences.

On the Soliton Solutions for the Stochastic Konno–Oono System in Magnetic Field with the Presence of Noise

In this study, we consider the stochastic Konno–Oono system to investigate the soliton solutions under the multiplicative sense. The multiplicative noise is considered firstly in the Stratonovich sense and secondly in the Ito^ sense. Applications of the Konno–Oono system include current-fed strings interacting with an external magnetic field. The F-expansion method is used to find the different types of soliton solutions in the form of dark, singular, complex dark, combo, solitary, periodic, mixed periodic, and rational functions. These solutions are applicable in the magnetic field when we study it at the micro level. Additionally, the absolute, real, and imaginary physical representations in three dimensions and the corresponding contour plots of some solutions are drawn in the sense of noise by the different choices of parameters.

Obtaining the Soliton Type Solutions of the Conformable Time-Fractional Complex Ginzburg–Landau Equation with Kerr Law Nonlinearity by Using Two Kinds of Kudryashov Methods

The main idea of this study is to obtain the soliton-type solutions of the conformable time-fractional complex Ginzburg–Landau equation with Kerr law nonlinearity. For this aim, the generalized and modified Kudryashov methods are applied to the given model. The reason for using a conformable derivative is that the chain rule can be applied to this derivative. Thus, using the suitable wave transform, the given equation is converted into an ordinary differential equation. Then, the proposed methods are applied to the reduced equation. According to our results, both of the used methods are effective and powerful. Finally, 3D and contour plots are given for some results with suitable variables. Our findings in this paper are critical for explaining a wide range of scientific and physical applications. According to our knowledge, our results are new in the literature.

A splitting lattice Boltzmann scheme for (2+1)-dimensional soliton solutions of the Kadomtsev-Petviashvili equation

<abstract> <p>Recently, considerable attention has been given to (2+1)-dimensional Kadomtsev-Petviashvili equations due to their extensive applications in solitons that widely exist in nonlinear science. Therefore, developing a reliable numerical algorithm for the Kadomtsev-Petviashvili equations is crucial. The lattice Boltzmann method, which has been an efficient simulation method in the last three decades, is a promising technique for solving Kadomtsev-Petviashvili equations. However, the traditional higher-order moment lattice Boltzmann model for the Kadomtsev-Petviashvili equations suffers from low accuracy because of error accumulation. To overcome this shortcoming, a splitting lattice Boltzmann scheme for (2+1)-dimensional Kadomtsev-Petviashvili-Ⅰ type equations is proposed in this paper. The variable substitution method is applied to transform the Kadomtsev-Petviashvili-Ⅰ type equation into two macroscopic equations. Two sets of distribution functions are employed to construct these two macroscopic equations. Moreover, three types of soliton solutions are numerically simulated by this algorithm. The numerical results imply that the splitting lattice Boltzmann schemes have an advantage over the traditional high-order moment lattice Boltzmann model in simulating the Kadomtsev-Petviashvili-Ⅰ type equations.</p> </abstract>

Degenerate and non-degenerate vector solitons and their interactions in the two-component long-wave–short-wave model of Newell type

The focus of this paper is on the dynamics of degenerate and non-degenerate vector solitons and their collision scenarios in the two-component long-wave–short-wave (2-LS) model of Newell type. The model equation that we study here governs the resonance interactions between two capillary waves (short waves) with a common gravity wave (long wave) for the surface wave of deep water. These degenerate and non-degenerate soliton solutions of the 2-LS model of Newell type are constructed via the bilinear KP-hierarchy reduction method and are expressed in forms of determinants. The degenerate solitons having only single-hump or single-valley profiles can undergo both elastic and inelastic collisions. Two particular types of degenerate solitons are also obtained, i.e., bound state solitons and V/Y-shaped solitons. The non-degenerate solitons and their collision scenarios have more interesting properties in contrast with the degenerate ones. The obvious one is that the non-degenerate one solitons possess symmetric or asymmetric double-hump and double-valley profiles if they propagate with identical velocities in all short-wave (SW) and long-wave (LW) components. Additionally, the soliton wavenumbers in the SW components and the LW component are unequal if the ones in the two SW components propagate with different velocities. The non-degenerate two solitons can undergo two different types of elastic collision scenarios in both SW and LW components: shape-preserving and shape-changing scenarios. We also study the collisions of non-degenerate solitons with degenerate ones, and find some intriguing properties. For instance, if the degenerate one soliton in one of the two SW components vanishes into background, then there is only one fundamental non-degenerate soliton in the corresponding SW component. However, this fundamental non-degenerate soliton still alters its profile from a symmetric double-hump waveform to an asymmetric double-hump one during its propagation process without interacting with other solitons. Furthermore, it is found a new class of soliton solutions referred to as (1,2,3) solitons. In these new types of soliton solutions, one SW component has only a single soliton, the other SW component has two solitons, whereas the LW component has three solitons.

Lie symmetry analysis, optimal system and exact solutions of variable-coefficients Sakovich equation

In this paper, the Sakovich equation is extended for the first time to a new equation with variable-coefficients on the time variable. The infinitesimal generators are obtained by performing a Lie symmetry analysis of the equation. Following this, the optimal system of one-dimensional subalgebras of the equation is analyzed. The (2+1)-dimensional variable-coefficients Sakovich equation is then reduced to (1+1)-dimensional partial differential equations by similarity reductions. The reduced partial differential equations are simplified to ordinary differential equations by the traveling wave transform. The new periodic wave solution and two-soliton interaction solution are obtained, and some exact solutions are obtained using the extended tanh-function method and the extended sech-function method. Finally, different types of soliton solutions are obtained by assigning different functions to the coefficient functions, and 3-dimensional figures are used to visualize the structural features of the soliton solutions.

Nonclassical potential symmetry analysis and exact solutions for a thin film model of a perfectly soluble anti-surfactant solution

In this article, we construct exact solutions to a quasilinear system of hyperbolic partial differential equations which governs the dynamics of a thin film of a perfectly soluble anti-surfactant solution. The symmetry analysis for this system is performed first time in the literature to the best of our knowledge. In fact, we present a detailed and comprehensive study of symmetry analysis for the governing system and compute classical symmetries, nonclassical symmetries, nonlocal symmetries and nonclassical potential symmetries. Nonclassical potential symmetries appear to be very much useful in terms of obtaining several new hidden solutions of the system that can not be established by using classical symmetry reductions, nonclassical symmetry method, or potential symmetry analysis. By using the direct multipliers we demonstrate several conserved quantities of the governing system those yield associated nonlocally related potential systems to the given system. Further, we prove that the given system admits a nonlocal symmetry that arises from the symmetry-based method and consequently we obtain a family of exact solutions. We analyze the physical interpretation of some of the obtained solutions which include various soliton type solutions such as kink type solitons, breather type solitons, multiple solitons, singular kink type solitons and etc. Lastly, as an application, we analyze the evolutionary property of characteristic shock, weak discontinuity and collision between them by using one of the obtained solutions.

Statistical properties of extreme soliton collisions.

Synchronous collisions between a large number of solitons are considered in the context of a statistical description. It is shown that, during the interaction of solitons of the same signs, the wave field is effectively smoothed out. When the number of solitons increases and the sequence of their amplitudes decay slower, the focused wave becomes even smoother and the statistical moments get frozen for a long time. This quasi-stationary state is characterized by greatly reduced statistical moments and by the density of solitons close to some critical value. This state may be treated as the small-dispersion limit, what makes it possible to analytically estimate all high-order statistical moments. While the focus of the study is made on the Korteweg-de Vries equation and its modified version, a much broader applicability of the results to equations that support soliton-type solutions is discussed.

Soliton Solutions and Sensitive Analysis of Modified Equal-Width Equation Using Fractional Operators

In this manuscript, the novel auxiliary equation methodology (NAEM) is employed to scrutinize various forms of solitary wave solutions for the modified equal-width wave (MEW) equation. M-truncated along with Atangana–Baleanu (AB)-fractional derivatives are employed to study the soliton solutions of the problem. The fractional MEW equations are important for describing hydro-magnetic waves in cold plasma. A comparative analysis is utilized to study the influence of the fractional parameter on the generated solutions. Secured solutions include bright, dark, singular, periodic and many other types of soliton solutions. In compared to other methods, the solutions demonstrate that the proposed technique is particularly effective, straightforward, and trustworthy that contains families of solutions. In addition, the symbolic soft computation is used to verify the obtained solutions. Finally, the system is subjected to a sensitive analysis. Integer-order results calculated by the symmetry method present in the literature can be addressed as limiting cases of the present study.

Two New Modifications of the Exp-Function Method for Solving the Fractional-Order Hirota-Satsuma Coupled KdV

Two novel modifications of the exp-function method have been suggested to solve the nonlinear system of the partial differential equation of the fractional order for the first time. The fractional-order Hirota-Satsuma coupled KdV system has been solved precisely; as a result, some exact solutions, which include soliton-type and rational solutions, will be derived. Eventually, the graphs of the obtained results have been illustrated at the end of the article. The newly used methods are highly accurate, flexible, effective, and programmable to solve fractional-order devices.

The Lie symmetry analysis and dynamics of exact solutions for variable coefficients (2+1)-dimensional coupled burgers equation

In this article, the (2+1)-dimensional variable coefficients coupled Burgers equation (vcCBE) is investigated for the first time with the help of Lie symmetry analysis method. This equation is an important nonlinear physical model. The optimal system of the (2+1)-dimensional vcCBE is analyzed by Olver’s method. Then, the (2+1)-dimensional vcCBE is reduced on the basis of the optimal system to multiple sets of (1+1)-dimensional equations. Various types of soliton solutions are obtained by solving the reduced equations. The -expansion method, tanh-coth method, and Riccati equation method are used respectively. Finally, the obtained solutions are analyzed dynamically and their kinetic behavior is investigated.

Abundant soliton-type solutions to the new generalized KdV equation via auto-Bäcklund transformations and extended transformed rational function technique

Abundant soliton-type solutions to the new generalized KdV equation via auto-Bäcklund transformations and extended transformed rational function technique

A novel approach to study generalized coupled cubic Schrödinger–Korteweg-de Vries equations

The Kortewegde Vries (KdV) equation represents the propagation of long waves in dispersive media, whereas the cubic nonlinear Schrödinger (CNLS) equation depicts the dynamics of narrow-bandwidth wave packets consisting of short dispersive waves. A model that couples these two equations seems intriguing for simulating the interaction of long and short waves, which is important in many domains of applied sciences and engineering, and such a system has been investigated in recent decades. This work uses a modified Sardar sub-equation procedure to secure the soliton-type solutions of the generalized cubic nonlinear Schrödinger–Korteweg-de Vries system of equations. For various selections of arbitrary parameters in these solutions, the dynamic properties of some acquired solutions are represented graphically and analyzed. In particular, the dynamics of the bright solitons, dark solitons, mixed bright-dark solitons, W-shaped solitons, M-shaped solitons, periodic waves, and other soliton-type solutions. Our results demonstrated that the proposed technique is highly efficient and effective for the aforementioned problems, as well as other nonlinear problems that may arise in the fields of mathematical physics and engineering.

Comparison between three distinct perceptions to the new solitary solutions of the generalized Hirota–Satsuma coupling KDV system

In this paper, we will implement a comparison between three different perceptions to the behavior of the exact solution to the generalized Hirota–Satsuma coupling Korteweg–de Vries System (GHSCKDVS). The proposed model plays a vital role in different branches of physics and applied mathematics. Specially, this model describes dispersive waves as the shallow water waves in hydrodynamics and it has various types of solutions like soliton solutions and traveling wave solutions. These three various visions to the behavior of the exact solution are established via three various techniques that have three different mathematical analyses. These three important different techniques are the Paul–Painlevé approach method (PPAM) examined previously to extract the exact solutions for many nonlinear partial differential equations (NLPDE) and achieved good results, the Ricatti–Bernolli Sub-ODE method (RBSODM) which is one of the famous ansatze approaches that doesn’t surrender to the balance rule, treats the balance rule fails and realizes impressive forms of the exact solutions. Many new various types of soliton solutions as hyperbolic and trigonometric function solutions, represented by the M-shaped, W-shaped, kink, anti-kink and dark soliton solutions have been established via these two methods. Moreover, we will document the identical numerical solutions for all achieved exact solutions realized by the two methods mentioned above via the famous numerical variational iteration method (VIM) that usually gives good results. The powerful of our achieved numerical solutions related to that their initial conditions emerged from the extracted exact solutions.

Charge self-trapping in two strand biomolecules: Adiabatic polaron approach

We investigate the properties of the excess charge (electron, hole) introduced into a two-strand biomolecule. We consider the possibility that the stable soliton excitation can be formed due to interaction of excess charge with the phonon subsystem. The influence of overlap of the molecular orbitals between adjacent structure elements of the macromolecular chain on the soliton properties is discussed. Special attention is paid to the influence of the overlapping of the molecular orbitals between structure elements placed on the different chains. Using the literature values of the basic energy parameters of the two-chain biomolecular structures, possible types of soliton solutions are discussed.

Novel soliton solutions of Sasa–Satsuma model with local derivative via an analytical technique

In this research article, the Sardar subequation method is used to retrieve new analytical solutions to the space-time local derivative Sasa–Satsuma equation with Atangana’s conformable derivative, which defines short pulse propagation in an optical fiber area. This equation is the integrable extension of the nonlinear Schrödinger equation. First, the equation is transformed into an ordinary differential equation utilizing traveling wave transformation. Then, novel different type soliton solutions are acquired using the Sardar subequation approach. The produced soliton solutions play an essential role for scientists in interpreting the physical phenomenon of this equation. Finally, the graphs of some solutions are depicted at appropriate values of parameters. The achieved results show the simplicity, reliability, and potentiality of the proposed method.

SOLITARY WAVES OF THE VARIANT BOUSSINESQ–BURGERS EQUATION IN A FRACTAL-DIMENSIONAL SPACE

In this work, we mainly focus on the fractal variant Boussinesq–Burgers equation which can well describe the motion of shallow water traveling along an unsmooth boundary. First, we construct its fractal variational principle and prove its strong minimum condition by the fractal Weierstrass theorem. Then two types of soliton solutions are acquired according to the constructed fractal variational principle. We find that the order of the fractal derivative hardly affects the whole shape of the solitary waves, but it remarkably affects its propagation process.