Soliton Research Articles

New analytical wave solitons and some other wave solutions of truncated M-fractional LPD equation along parabolic law of non-linearity

New analytical wave solitons and some other wave solutions of truncated M-fractional LPD equation along parabolic law of non-linearity

Nonlinear superposition between lump soliton and other nonlinear localized waves for the (2+1)-dimensional Korteweg–de Vries–Sawada–Kotera–Ramani equation

In the previous studies, the authors have reported the lump soliton collided with other nonlinear localized waves for (2+1)-dimensional Korteweg–de Vries–Sawada-Kotera–Ramani equation. However, this work will introduce a novel restrictive condition to reveal the nonlinear superposition between a lump soliton and other nonlinear localized waves. Under the premise of the acquired N-soliton solutions, the nonlinear superposition structures among lump soliton, stripe solitons, resonance Y-type solitons and breather waves guarantee that the lump soliton does not collide with other localized waves by means of the long wave limit approach, soliton resonance method and symbolic computation. Meanwhile, a variety of soliton molecules are yielded through nonlinear superposition and velocity resonance mechanisms. The obtained solutions in this work will bring about new perception of the interaction behavior between lump soliton and other nonlinear localized waves, and provide more brand new insights into the nonlinear phenomena triggered by the areas of plasma physics, fluid dynamics and nonlinear optics.

Hodograph transformation, various exact solutions and dynamical analysis for the complex Wadati–Konno–Ichikawa-II equation

This paper is devoted to studying the second type of complete integrable complex Wadati–Konno–Ichikawa-II (cWKI-II) equation. We first study the conservation laws and construct the hodograph transformation for this equation. Based on the resulting transformation, we construct a new two-component associated cWKI-II equation, which is completely equivalent to the cWKI-II equation. Based on the corresponding Lax pair for this new associated equation, we construct its N-fold Darboux transformation and generalized (m,N−m)-fold Darboux transformation. Then, the associated Bäcklund transformation is used to give various exact solutions, such as smooth soliton, bursting soliton, singular-loop soliton, position controllable semi-rational soliton, breather and rogue wave solutions. The soliton structures along with dynamical evolution behaviors are discussed and shown graphically. Moreover, we investigate some important physical quantities of some soliton solutions such as amplitude, wave number, wave width, velocity, energy and phase by using the asymptotic analysis technique. Finally, we also summarize the formation reasons for some singular structures under some parameter constraints.

Single and Bunch Soliton Generation in Optical Fiber Lasers Using Bismuth Selenide Topological Insulator Saturable Absorber.

In this work, we present the generation of two distinct types of soliton pulses using a Bismuth Selenide (Bi2Se3) saturable absorber (SA) synthesized in our laboratory. The soliton pulses were generated in two different laser cavity configurations, resulting in two types of solitons: a soliton pulse with Kelly sidebands and a bunched soliton pulse with peak-dip sidebands. Both solitons operated at the fundamental repetition rate-23.3 MHz (for the soliton with Kelly sidebands) and 13 MHz (for the bunched soliton with peak-dip sidebands). We observed that the accumulation of nonlinear phase shift from the added single mode fiber (SMF) split the single soliton pulse into 44 pulses in a bunched oscillation envelope. At the same time, peak-dip sidebands were imposed on the bunched soliton spectrum due to constructive and destructive interferences between soliton pulse and dispersive waves. The measured pulse width for both solitons were 0.63 ps (for the soliton with Kelly sidebands) and 1.52 ps (for the bunched soliton with peak-dip sidebands), respectively. Our results demonstrate the potential of Bi2Se3 SAs in generating different types of soliton pulses, which could have potential applications in various areas of optical communication and spectroscopy.

Prime Numbers Among Fractional, Integer and Natural Numbers

Series of primes are considered in three levels of sets of numbers (N)(Z)(Q), where – series of prime numbers in the set of natural numbers (PN), – a series of integer primes in the set of integers (WPN), – a series of rational primes in the set of rational (fractional) numbers (FPN). In series and the abscissa is taken as a complete series of integers , and for a series of prime numbers the positive right side of the x-axis is taken as a series of natural numbers . It is proved that the FPN set contains four types in the quadrants of the Cartesian coordinate system, the WPN set includes two mirror-opposite lines (the left one in quadrants II and IV and the right one in quadrants I and III). Subtraction from the next term WPN and PN of the previous component of the series gives new series of increments. Converting from decimal to binary gives for the series WPN and PN, as well as their increments, the rational root 1/2 of the Riemann zeta function on the second vertical of the binary expansion. Such a transformation makes it possible to obtain the block structure WPN and PN of partitioning the original series into blocks, and then the groups starting from the number 2. Using the example of block No. 8168 for the series PN, we proved the decomposition of groups into separate quanta, starting from the number 2, by asymmetric wavelets in the form solitary waves (solitons) with variable amplitude and oscillation period with a maximum relative error of 0.1%.

Abundant Resonant Behaviors of Soliton Solutions to the (3+1)-dimensional BKP-Boussinesq Equation

Resonant phenomena have been observed and investigated in various situations, such as plasma experiments, the maritime security and the microtubule in cell physiology. In this paper, abundant resonant behaviors are studied for the (3+1)-dimensional BKP-Boussinesq equation. We mainly discuss the resonant two- and three-soliton solutions in the (x, y)-plane and (x, z)-plane. The characteristics are given for the kink soliton waves, including expressions, maximums, minimums and velocities. The kink soliton waves in the (x, y)-plane are parallel, and the fusion or fission may occur. The kink soliton waves in the (x, z)-plane are not parallel and the resonant phenomena among them are more complicated.

New optical soliton solutions for time-fractional Kudryashov’s equation in optical fiber

In this paper, the new Kudryashov approach is considered to find several new exact solutions to time-fractional Kudryashov’s equation which describes the propagation pulses in optical fibers. The present novel optical solutions are expressed via the exponential and hyperbolic functions which are categorized as dark-bright, dark, bright, bell-shape, wave, and singular optical soliton solutions. The two-dimension, three-dimension, and cantor graphs of the dark, bright, bell-shape, wave, and singular solutions are depicted to illustrate the magnitude of the time-fractional Kudryashov’s equation choosing suitable values of physical parameters. Furthermore, the effect of the parameter of time and the conformable fractional order derivative is also presented. Finally, the new Kudryashov approach is an efficient technique to analysis the analytic solutions of the differential equations of integer and fractional orders.

A new implementation of a novel analytical method for finding the analytical solutions of the (2+1)-dimensional KP-BBM equation

In this work, we perform a comprehensive analytical study to find the novel exact traveling wave solutions of the (2+1)-dimensional Kadomtsev-Petviashvili-Benjamin-Bona-Mahony (KP-BBM) equation. The recently developed (G′G′+G+A) -expansion technique is a capable method for finding the new exact solutions of assorted nonlinear evolution equations. Some new analytical solutions are obtained by utilizing the aforementioned method. The obtained solutions are expressed as trigonometric functions and exponential functions. The extracted exact wave solutions are advanced and fully unique from the earlier literature Moreover, we have presented the contour simulations, 2D and 3D graphical representations of the solution functions and we have observed that the solutions obtained are periodic and solitary wave solutions. We have shown two soliton wave solutions and two singular periodic wave solutions for the particular values of the parameters graphically. As per our knowledge, we must say that the extracted solutions might be significant and essential for new physical phenomenon.

Dynamic study of bifurcation, chaotic behavior and multi-soliton profiles for the system of shallow water wave equations with their stability

The purpose of this study is to investigate the deeper characteristics of the system of shallow water wave equations that is used to model the turbulence in the atmosphere and oceans from different standpoints. The multi-soliton structures such as 1-soliton, 2-soliton, 3-soliton solutions are successfully generated with the help of simplified Hirota’s method. For the sake of physical demonstration and visual presentation, we graphically illustrate the identified solutions in 3D, 2D, and contour plots using Mathematica software. The Lie symmetry technique is used to identify the Lie group invariant transformations and symmetry reductions of the examined system, which aid in reducing the dimension by one or into an ordinary differential equation. Further, the qualitative behavior of the time-dependent dynamical system is observed using the bifurcation and chaos theory. The phase portraits of bifurcation are observed at the equilibrium point of a planar dynamical system. We discuss various tools to identify chaos (random and unpredictable behavior) in autonomous dynamic systems, such as 3D phase portraits, 2D phase portraits, time series, and Poincaré maps. At various initial conditions, the sensitivity and modulation instability analyses are also presented, and it is discovered that the investigated system is stable, as a small change in the initial conditions does not cause an abrupt change in solutions. The findings of this investigation will contribute in the overall depiction of soliton theory and nonlinear dynamical systems.

Temporal behavior of bright and dark spatial solitons in photorefractive crystals having both the linear and quadratic electro-optic effects based on low amplitude approximations

We investigate the formation or time evolution characteristics of bright and dark optical spatial solitons in novel photorefractive media with linear and quadratic electro-optic effects. This study was carried out at low amplitude. By the application of low amplitude approximation to the time-dependent wave (soliton) propagation equation, bright soliton solution (BSS) and dark soliton solution (DSS) were each obtained analytically, and both are exact solutions. The change in soliton width or the intensity full width at half maximum (FWHM) of two soliton types can also be calculated directly from the BSS and the DSS, providing a suitable description of the bright and dark solitons time evolution in novel photorefractive media. Only steady state solitons are formed in low amplitude, and the quasi-steady state soliton does not exist as reported at high saturation values. We compare analytical BSS with numerical calculation results of previous studies and find good agreement between analytics and numerics on the amplitude limit. In contrast to bright solitons, which can arise in the dominant linear or quadratic electro-optic effect, dark solitons in novel photorefractive media always exist under the dominant quadratic electro-optic effects only. We found that the time evolution of dark solitons from the DSS corresponds with time-independent optical spatial solitons studies in a similar media. Lastly, BSS and DSS found at low amplitude limits are fundamental soliton solutions or initial state soliton (at low peak intensity ratio regime) that can play an important role in further investigations of optical spatial solitons formation in novel photorefractive media.

KdV breathers on a cnoidal wave background

Using the Darboux transformation for the Korteweg–de Vries equation, we construct and analyze exact solutions describing the interaction of a solitary wave and a traveling cnoidal wave. Due to their unsteady, wavepacket-like character, these wave patterns are referred to as breathers. Both elevation (bright) and depression (dark) breather solutions are obtained. The nonlinear dispersion relations demonstrate that the bright (dark) breathers propagate faster (slower) than the background cnoidal wave. Two-soliton solutions are obtained in the limit of degeneration of the cnoidal wave. In the small amplitude regime, the dark breathers are accurately approximated by dark soliton solutions of the nonlinear Schrödinger equation. These results provide insight into recent experiments on soliton-dispersive shock wave interactions and soliton gases.

Exact solutions and Darboux transformation for the reverse space–time nonlocal Lakshmanan–Porsezian–Daniel equation

The integrable reverse space–time nonlocal Lakshmanan–Porsezian–Daniel (LPD) equation is proposed by nonlocal reduction of the AKNS scattering problem. The 1-fold and N-fold Darboux transformations of the nonlocal LPD equation are constructed, from which some interesting nonlinear wave solutions including soliton and rogue wave solutions are obtained, whose dynamical properties are discussed by evolution plots. It is worth mentioning that the real and imaginary parts of the soliton solutions show the characteristics of the breathing solution. These results may be useful in nonlinear optics and related fields.

Breather and rogue wave solutions for the generalized discrete Hirota equation via Darboux–Bäcklund transformation

This paper investigates the Darboux–Bäcklund transformation, breather and rogue wave solutions for the generalized discrete Hirota equation. The pseudopotential of this equation is proposed for the first time, from which a Darboux–Bäcklund transformation is constructed. Starting from a more general plane wave solution and applying the obtained transformation, a variety of nonlinear wave solutions, including three types of breathers, W-shaped soliton, periodic solution and rogue wave are obtained, and their dynamical properties and evolutions are illustrated by plotting figures. The relationship between parameters and wave structures is discussed in detail. The method and technique employed in this paper can also be extended to other nonlinear integrable equations. Our results may help us better understand some physical phenomena in optical fibers and relevant fields.

Various Solitons and Other Wave Solutions to the (2+1)-Dimensional Heisenberg Ferromagnetic Spin Chain Dynamical Model

This paper outlines a study into the exact solutions of the (2+1)-dimensional Heisenberg ferromagnetic spin chain equation that is used to illustrate the ferromagnetic materials of magnetic ordering by applying two recent techniques, namely, the Sardar-subequation method and extended rational sine–cosine and sinh–cosh methods. Abundant exact solutions such as the bright soliton, dark soliton, combined bright–dark soliton, singular soliton and other periodic wave solutions expressed by the generalized trigonometric, generalized hyperbolic, trigonometric and hyperbolic functions are obtained. The numerical results are illustrated in the form of 3D plots, 2D contours and 2D curves by choosing proper parametric values to interpret the physical behavior of the model. The obtained results in this work are expected to provide a rich platform for constructing the soliton solutions of PDEs in physics.

Rational and semi-rational solution to the (3+1)-dimensional Kadomtsev-Petviashvili-Boussinesq-like equation

In the paper, we take the (3+1)-dimensional KP-Boussinesq-like equation as an instance and utilize long wave limit approach to generate rational solution and semi-rational solution from the N-order soliton solution. For N = 2, 3, with specific parametric constraints, we have constructed rational solutions which can be presented as first order line rogue wave and single breather from two-soliton solution, and semi-rational solution is classified as first order kink-shaped rogue wave and a hybrid of breather and one soliton from three-soliton solution. For N = 4, the rational solution includes second order line rogue wave, double breather, interaction between two breathers and a hybrid of breather and line rogue wave. The semi-rational solution contains second order kink-shaped rogue wave and interaction between breather, soliton and line rogue wave. In addition, we give mathematical expressions for higher N-order rational solution.

Exact solutions and Darboux transformation for the reverse space–time non-local fifth-order non-linear Schrödinger equation

In this paper, the non-local reverse space−time fifth-order non-linear Schrödinger(NLS) equation has been investigated, which is proposed by the non-local reduction of Ablowitz–Kaup–Newell–Segur (AKNS) scattering problems. The determinant representation of the Nth Darboux transformation for the non-local reverse space−time fifth-order NLS equation is obtained. Some interesting non-linear wave solutions, including soliton, complexiton, and rogue wave solutions, are derived by the Darboux transformation. Moreover, the dynamics of non-linear wave solutions are illustrated with the corresponding evolution plots, and the results show that the non-local fifth-order NLS equation has new different properties from the local case.

Solitary wave solution of (2+1)-dimensional Chaffee–Infante equation using the modified Khater method

Most of the nonlinear phenomena are described by partial differential equation in natural and applied sciences such as fluid dynamics, plasma physics, solid state physics, optical fibers, acoustics, biology and mathematical finance. The solutions of a wide range of nonlinear evolution equations exhibit the wave behavior corresponding to the underlying physical systems. In particular, solitary wave solutions and soliton solutions are of great interest for researchers owing to many applications in different areas of science. The behavior of gas diffusion in a homogeneous medium is described by the (2+1)-dimensional Chaffee–Infante equation. In the present study, the modified Khater method is used to solve (2+1)-dimensional Chaffee–Infante equation because it provides various forms of solitons. The bright, dark, and periodic traveling wave patterns are produced by choosing different values of parameters. The solutions are presented graphically through 2D, 3D and contour graphs. The obtained solutions are demonstrated that the modified Khater method is a more useful tool than the existing techniques for solving such nonlinear problems.

Study of compressive and rarefactive lump solitons structures in dusty plasma with double spectral (r,q) distributed electrons

Nonlinear features are revealed by one of the most competent and powerful approaches, i.e. soliton theory. The present article starts with the dust collisionless magnetized plasma where electrons are following double spectral ( r , q ) distribution. From the hydrodynamical governing equations set, Kadomtsev–Petviashvili (KP) equation is derived using the reductive perturbation technique. The derived KP equations are converted to standard KP equations by a suitable transformation to use the Hirota bilinear method (HBM) more conveniently. By using HBM, an auxiliary function is constructed, and through symbolic computation, three arbitrary constant sets are formed. These sets associate with three lump soliton solution sets. Finally, all these lump soliton solutions are returned back by the inverse transformation that was used to make the derived KP equation to the standard KP equation and get the lump soliton solution of the associated system. It is investigated that associated plasma parameters have a significant impact on the lump soliton structures while tracing figures using these plasma parameters within the justified range of the system. While lump soliton features' are analysed for various parameters' effect on it, one most important aspect is revealed. Double spectral index r and q play a significant role in the lump solitons structures and depict compressive and rarefactive lump soliton structures with the associated system. It is found that these double spectral indices r and q are the decision-making parameters for the lump soliton structures to be compressive or rarefactive.

Analytical solitary wave solutions of a time-fractional thin-film ferroelectric material equation involving beta-derivative using modified auxiliary equation method

The main objective of current paper is to examine the impacts of fractional parameters on the dynamic response of soliton waves in a nonlinear time-fractional thin-film ferroelectric materials equation (TFFEME). To achieve such a goal, the TFFEME is first rehabilitated into ordinary differential equation using a complex wave transformation. Solitary wave solutions of the governing equation, representing the dynamics of waves in the material and plays a vital rule in many branches of physics and hydrodynamics, are then constructed through applying the modified auxiliary equation method (MAEM). The extracted solutions are demonstrated using definition of the beta derivative to understand their dynamical behavior. The hyperbolic, periodic and trigonometric function solutions are used to derive the analytical solutions for the given model. As a result, dark, bright, periodic and solitary wave solitons are obtained. The fractional impact of the above derivative on the physical phenomena is observed. The 2D and 3D graphs are also shown to confirm the behavior of analytical wave solutions.

Multiple soliton and M-lump waves to a generalized B-type Kadomtsev–Petviashvili equation

In this study, we focus on the (3+1)-dimensional generalized B-type Kadomtsev–Petviashvili (gBKP) equation in fluid dynamics, which is useful for modeling weakly dispersive waves transmitted in quasi media and fluid mechanics. As a general matter, this paper examines the gBKP equation including variable coefficients of time that are widely employed in plasma physics, marine engineering, ocean physics, and nonlinear sciences to explain shallow water waves. Using Hirota’s bilinear approach, one-, two, and three-soliton solutions to the problem are constructed. By employing a long-wave method, 1-M-, 2-M, and 3-M-lump solutions are derived. In addition, interaction phenomena of one-, and two-soliton solutions with one-M-lump wave are revealed. Moreover, an interaction solution between a two-M-lump wave and a one-soliton solution is also offered. The planes that M-lump waves travel among them are derived. We believe that our findings will help improve the dynamical properties of (3+1)-dimensional BKP-type equation.