Soliton Research Articles

Numerical solution of KdV equation in dusty plasma system using Fourier transform

We consider dusty plasma in a cometary environment comprising of positively and negatively charged dust ions, kappa distributed - solar electrons, cometary electrons and hydrogen ions. The existence of non linear waves such as solitons in these systems were explored in detail by various researchers analytically. In this article we solve the system numerically by deriving KdV equation and solving it using Fourier transform. Hence we study the generation and existence of solitons. We also explore the characteristics of the solitons formed by simulation of the system for various initial conditions. The system is found to have single soliton wave as exact solution and set of soliton wave train solutions for varied initial conditions. The soliton wave trains can be compressive, rarefactive or mixed in nature according to the initial condition. The simulation is helpful in understanding and modelling various dusty plasma systems.

Soliton–mean field interaction in Korteweg–de Vries dispersive hydrodynamics

AbstractThe mathematical description of localized solitons in the presence of large‐scale waves is a fundamental problem in nonlinear science, with applications in fluid dynamics, nonlinear optics, and condensed matter physics. Here, the evolution of a soliton as it interacts with a rarefaction wave or a dispersive shock wave, examples of slowly varying and rapidly oscillating dispersive mean fields, for the Korteweg–de Vries equation is studied. Step boundary conditions give rise to either a rarefaction wave (step up) or a dispersive shock wave (step down). When a soliton interacts with one of these mean fields, it can either transmit through (tunnel) or become embedded (trapped) inside, depending on its initial amplitude and position. A topical review of three separate analytical approaches is undertaken to describe these interactions. First, a basic soliton perturbation theory is introduced that is found to capture the solution dynamics for soliton–rarefaction wave interaction in the small dispersion limit. Next, multiphase Whitham modulation theory and its finite‐gap description are used to describe soliton–rarefaction wave and soliton–dispersive shock wave interactions. Lastly, a spectral description and an exact solution of the initial value problem is obtained through the inverse scattering transform. For transmitted solitons, far‐field asymptotics reveal the soliton phase shift through either type of wave mentioned above. In the trapped case, there is no proper eigenvalue in the spectral description, implying that the evolution does not involve a proper soliton solution. These approaches are consistent, agree with direct numerical simulation, and accurately describe different aspects of solitary wave–mean field interaction.

Dispersive optical solitons with DWDM topology and multiplicative white noise

The primary objective of this research is to investigate and analyze the intricate dynamics exhibited by dispersive optical soliton solutions within dispersion-flattened fibers while considering the impact of white noise. By examining this phenomenon, the study aims to gain a comprehensive understanding of the behavior and characteristics of these solitons, which are essential in the field of optical communication. A wide range of soliton solutions is obtained by the extended auxiliary equation procedure and the unified Riccati equation scheme, which are two powerful methods employed in the field of nonlinear dynamics and soliton theory to obtain a diverse array of soliton solutions. These techniques enable researchers to explore and analyze a wide range of complex nonlinear phenomena in various physical systems. This wide range of soliton solutions contributes to a deeper understanding of nonlinear phenomena and provides valuable insights into the fundamental properties and applications of solitons in different areas of science and engineering, including optics, fluid dynamics, plasma physics, and condensed matter physics. White noise only affects the phase part of the solitons and does not spread to other components. Such conclusions are applicable only under the specific transformation and the special simplification employed in deriving the solutions in the paper. The existence of white noise does not change the soliton amplitudes, widths, or velocities in any component. This paper is the first to report an essential observation concerning dispersion-flattened fibers, which has never been reported before.

Modulation instability gain and localized waves in the modified Frenkel–Kontorova model with high-order nonlinearities

In this paper, modulation instability and nonlinear supratransmission are investigated in a one-dimensional chain of atoms using cubic–quartic nonlinearity coefficients. As a result, we establish the discrete nonlinear evolution equation by using the multi-scale scheme. To compute the modulation instability gain, the linearizing technique is employed. The effect of the higher nonlinear component on modulation instability is particularly examined. Following that, full numerical integration was performed to identify modulated wave patterns as well as the appearance of a rogue wave. Through the nonlinear supratransmission phenomenon, one end of the discrete model is driven into the forbidden bandgap. As a consequence, for driving amplitudes above the supratransmission threshold, the bright soliton and modulated wave patterns are satisfied. An important behavior is observed in the transient range of time of propagation when the bright soliton wave turns into a chaotic soliton wave. These results corroborate our analytical investigations on the modulation instability and show that the one-dimensional chain of atoms is a fruitful medium to generate long-lived modulated waves.

Electrostatic wave propagation and self-streaming effect in an electron-hole plasma

Electrostatic nonlinear waves which transfer energy through the semiconductor are investigated. A quantum hydrodynamic plasma system composed of self-streaming electrons and holes is examined. The basic equations are reduced to one evolution equation called a modified nonlinear Schrödinger (mNLS) equation. The stability and instability regions are studied with respect to the wavenumber and different plasma effects such as degenerate pressure, Bohm potential, and collisions. The mNLS equation is solved analytically to obtain three kinds of nonlinear envelope wave packet modes. It is found that there are different regions of stability and instability depending on various quantum effects. The electrons’ and holes’ self-streaming velocity is studied and manipulated for the three types of nonlinear envelope waves ‘dark soliton, bright soliton, and rogue wave’. The dark envelope wave packet is generated in a stable region. When the electrons and holes streaming velocities become faster, the wave amplitude becomes taller and the pulses have higher frequency. The bright envelope wave packet exists in the unstable region. For low streaming velocities, the rogue wave amplitude becomes shorter, however, when the streaming velocities reach a critical value the amplitude increases suddenly six times. The self-heating could be produced as the tunneling electrons and holes exchange their energy with the lattice, which may decrease the lifetime of the semiconductors. The present results are helpful in realizing the physical solution to the intrinsic heating problem in semiconductors.

Study of generalized regularized long wave equation via septic Hermite collocation method with Crank–Nicolson and SSP-RK43 schemes to capture the various solitons

The paper aims to develop a powerful numerical technique for the generalized regularized long wave equation. The proposed technique is based on the orthogonal collocation on finite element method and septic Hermite interpolation polynomials as basis functions called septic Hermite collocation method (SHCM). These basis functions are C3 continuous, ensuring the continuity of dependent variable and its first three derivatives throughout the solution range. Two schemes viz one is implicit Crank–Nicolson scheme and other is Runge–Kutta method of four stages and third-order (SSP-RK43) are used for time discretization and SHCM is used for space discretization. The stability analysis of SHCM with Crank–Nicolson is performed and the scheme is found to be unconditionally stable. The stability of SHCM with SSP-RK43 by CFL conditions is also studied and the method is found to be conditionally stable. The proposed methods are applied to three problems involving single solitary waves, the interaction of two solitary waves, and the evolution of solitons via the Maxwellian initial condition. The formation and behavior of soliton waves are studied numerically. To demonstrate the robustness of proposed technique, error norms and the invariants of motion, i.e., mass, momentum, and energy are calculated. The results are in good agreement with the exact ones and found to be better than the numerical solutions available in the literature. The order of convergence along spatial and temporal directions has been calculated numerically. It is observed that the order of convergence of the explicit SSP-RK43 is higher than the implicit Crank–Nicolson and SHCM with SSP-RK43 provides better accuracy than the SHCM with Crank–Nicolson. The SHCM is capable of displaying solitary wave collision. The proposed technique is simple, fast, efficient, and easy to implement. It does not require unnecessary integration or calculation of weight functions as the case of other numerical methods for instance Galerkin methods, spectral methods, B-spline finite element methods etc.

On the propagation of cnoidal wave and overtaking collision of slow shear Alfvén solitons in low β magnetized plasmas

The overtaking collisional phenomenon of slow shear Alfvén solitons are studied in a low beta (β = kinetic pressure/magnetic pressure) collisionless, magnetized plasma consisting of electron and ion fluids. By employing a reductive perturbation technique, the Korteweg–de Vries (KdV) equation is deduced for investigating the nonlinear slow shear Alfvén wave. Before embarking on the study of the overtaking collisions, the stability analysis of the KdV equation is studied using the bifurcation theory. Also, a nonlinear periodic solution of the KdV equation is derived for the first time in the Weierstrass elliptic function formula. Moreover, the condition for converting the Weierstrass elliptic function expression to soliton is discussed. Furthermore, it is found that only density dip (rarefactive) solitons are formed in the super-Alfvénic regime. The next step includes the use of the Hirota bilinear method, which results in two and three shear Alfvén soliton solutions and their subsequent phase shifts. The influence of the plasma parameters on the amplitude as well as width of the slow shear Alfvén wave solitons are examined analytically and numerically. We also find out the profiles of overtaking interaction of slow shear Alfvén dip solitons having different amplitudes and speeds numerically. This study is important for understanding the phenomena of nonlinear slow shear Alfvén wave structures both in space and in laboratory plasmas.

A family of solutions of the time–space fractional longitudinal wave equation

In this article, we have studied a nonlinear time–space fractional longitudinal wave equation in the context of the conformable fractional derivative. Through the soliton ansatz method and a direct integration approach with the symmetry condition, new soliton and solitary wave solutions are derived. Furthermore, the existing conditions of these obtained solutions are also given in this text. These new results add to the existing literature. We believe that they can provide a new window into the understanding of this model.

Propagation of Nonlinear Dust-Acoustic Waves in a Self-Gravitating Collision Magnetized Dusty Plasma in Earth’s Magnetosphere

The nonlinear propagation of different types of DANW acoustic dust nonlinear waves has been investigated in a magnetized dusty plasma consisting of negatively charged dust particles, Maxwellian electrons, and ions. Application of the standard reductive perturbation theory is used to derive the corresponding three-dimensional nonlinear a complex Ginzburg–Landau (3D-CGLE) equation which governs the dynamics of the dust-acoustic wave packets. The stationary analytical solutions of the CGLE are numerically analysed where the effect of the physical parameters of the dusty plasma model on the wave’s propagation is taken into account. It has been found that there can be a relationship between the appearance of soliton waves and electromagnetic waves, as well as between shock-like waves and periodic travelling waves. Expression of the importance of these findings is the cornerstone of explaining the true relationship between the propagation of nonlinear waves in the physics of space, for example, the Earth’s magnetic field.

Analytical and numerical investigation of soliton wave solutions in the fifth-order KdV equation within the KdV-KP framework

This study investigates the fifth-order Korteweg–de Vries (fKdV) equation, which is a generalized form of the classical KdV equation governing weakly nonlinear waves in dispersive media. By employing analytical and numerical techniques, the Khater II and variational iteration methods are utilized to obtain accurate soliton wave solutions. The combination of these approaches ensures the reliability and precision of the analytical outcomes. Graphical representations, including two-dimensional, three-dimensional, and contour plots, visually illustrate the results. Additionally, the stability of the constructed solutions is assessed through the characterization of the Hamiltonian system, providing valuable insights into the soliton wave solutions’ stability properties. This research integrates analytical and numerical methods to explore the fKdV equation within the KdV-KP framework, offering novel soliton wave solutions and enhancing their interpretation through graphical representations while assessing their stability using the Hamiltonian system’s characterization.

The Investigation of Dynamical Behavior of Benjamin–Bona–Mahony–Burger Equation with Different Differential Operators Using Two Analytical Approaches

The dynamic behavior variation of the Benjamin–Bona–Mahony–Burger (BBM-Burger) equation has been investigated in this paper. The modified auxiliary equation method (MAEM) and Ricatti–Bernoulli (RB) sub-ODE method, two of the most reliable and useful analytical approaches, are used to construct soliton solutions for the proposed model. We demonstrate some of the extracted solutions using definitions of the β-derivative, conformable derivative (CD), and M-truncated derivatives (M-TD) to understand their dynamic behavior. The hyperbolic and trigonometric functions are used to derive the analytical solutions for the given model. As a consequence, dark, bell-shaped, anti-bell, M-shaped, W-shaped, kink soliton, and solitary wave soliton solutions are obtained. We observe the fractional parameter impact of the derivatives on physical phenomena. The BBM-Burger equation is functional in describing the propagation of long unidirectional waves in many nonlinear diffusive systems. The 2D and 3D graphs have been presented to confirm the behavior of analytical wave solutions.

Exact traveling wave solutions and soliton solutions of conformable M-fractional modified nonlinear Schrödinger model

The traveling wave solutions and soliton solutions of the conformable M-fractional modified nonlinear Schrödinger equation have been devised in this work by employing the generalized projective Riccati equations technique. The proposed method is straightforward and concise to obtain exact soliton solutions of nonlinear partial differential equations in various scientific discipline. This approach has been applied successfully to a wide variety of physical and mathematical problems. In many cases, the method has been able to provide new insights into the behavior of these systems and has led to the discovery of new phenomena. Several types of soliton solutions, including hyperbolic traveling wave solution, kink-shape soliton solution, singular soliton solution and dark soliton-type traveling wave solution, have been derived from the governing equation by using this effective technique. The solutions achieved by the governing model contain significant and dynamic justifications for certain real-world physical phenomena. Some acquired soliton solutions are displayed in 3D plots, corresponding contour plots and density plots to illustrate the physical dynamics of the governing model. Furthermore, the resulting outcomes manifest that the proposed method is highly efficient for revealing wave solutions for a wide range of nonlinear partial differential equations that emerge in optics.

Dust acoustic solitary waves in dusty plasma with nonuniform temperature

In this paper, the dust acoustic wave is studied in conditions where the dust temperature is not constant. This model includes negatively charged dust particles and thermal ions. The reductive perturbation technique is used and the new point in this study is the nonuniformity of the plasma temperature. The nonconstancy of the temperature is entered as a first-order perturbation in the calculations, and finally the modified Korteweg–de Vries (mKdV) equation is derived. The solution of the modified KdV equation clarifies the change in soliton shape of the wave when it moves in the perturbation plasma. We show how the soliton wave undergoes deformation and amplitude reduction during propagation when it encounters a region with a different temperature. The output of this research can be effective to better understand the wave behavior in a real plasma with nonuniform temperature.

Two-photon imaging of soliton dynamics

Optical solitary waves (solitons) that interact in a nonlinear system can bind and form a structure similar to a molecule. The rich dynamics of this process have created a demand for rapid spectral characterization to deepen the understanding of soliton physics with many practical implications. Here, we demonstrate stroboscopic, two-photon imaging of soliton molecules (SM) with completely unsynchronized lasers, where the wavelength and bandwidth constraints are considerably eased compared to conventional imaging techniques. Two-photon detection enables the probe and tested oscillator to operate at completely different wavelengths, which permits mature near-infrared laser technology to be leveraged for rapid SM studies of emerging long-wavelength laser sources. As a demonstration, using a 1550 nm probe laser we image the behavior of soliton singlets across the 1800–2100 nm range, and capture the rich dynamics of evolving multiatomic SM. This technique may prove to be an essential, easy-to-implement diagnostic tool for detecting the presence of loosely-bound SM, which often remain unnoticed due to instrumental resolution or bandwidth limitations.

Exploring the dynamics of soliton waves: A comparative analysis of analytical and numerical methods for the modified Equal-Width equation

The modified Equal-Width (MEW) equation is a fundamental model in the realm of partial differential equations that is used to simulate one-dimensional wave propagation in nonlinear media, incorporating dispersion processes. In this study, we have employed two contemporary analytical and numerical methods to obtain exact traveling wave solutions for this model. The derived solutions have been evaluated using a numerical technique, demonstrating excellent accuracy. Various graphical representations, including a contour plot, a two-dimensional graph, and a three-dimensional graph, have been utilized to illustrate and analyze the results. Our study offers significant contributions to the field by providing new insights into the dynamics of solitary wave solutions of the MEW model. Furthermore, our method proves to be a valuable and effective mathematical tool that can be utilized to solve various nonlinear wave problems, making it highly applicable in a range of physical and mathematical studies. By showcasing the properties of the investigated model and the primary advantages of the employed techniques, our study highlights the potential for further developments and advancements in this field.

Interaction of two soliton waves in plasma including electrons with Kappa-Cairns distribution function

• The nonlinear propagation of soliton waves can be expressed by different equations. • KdV equation includes the equilibrium between nonlinearity and dispersion sentences, which gives information about the soliton waves. • One of the distribution functions that has been utilized in the last few years is the C-T distribution function. • The results demonstrated that the parameters and have a great impact on phase. velocity, nonlinear and dispersive coefficients, Sagdeev potential, maximum amplitude, and width of IASWs. a q The interaction of positron acoustic soliton waves (PASWs) with the arbitrary collision angle in plasma including cold fluid positrons, stationary ions and electrons with Kappa-Cairns (K-C) distribution function have been studied. The equations of Korteweg-de Vries (KdV) and the phase shifts are obtained by employing the extended Poincaré–Lighthill–Kuo (PLK) method for the two colliding waves. The influences of parameters of the K-C distribution function ( κ and α ), the collision angle θ and the proportion of the ion (electron) and positron unperturbed densities ( β i ( β e )) on the phase shifts are investigated.

Dust ion acoustic bi-soliton, soliton, and shock waves in unmagnetized plasma with Kaniadakis-distributed electrons in planar and nonplanar geometry

Dust ion acoustic bi-soliton, soliton, and shock waves in unmagnetized plasma with Kaniadakis-distributed electrons in planar and nonplanar geometry

Parametric effects on paraxial nonlinear Schrödinger equation in Kerr media

• The mentioned techniques are applied to the paraxial NLS equation in Kerr media. • Optical soliton solutions are obtained and has parameters connected to the model. • Analyzed the nature of the obtained solutions by 3D and 2D wave profiles. • Nonlinear and free parameters effects on the paraxial NLS equation have been analyzed. • Wave profiles streamline pattern and immediate local directions have been analyzed. In this study, we have considered the (2+1)-dimensional paraxial nonlinear Schrödinger (NLS) equation in Kerr media and used the ( w / g ) - expansion method. The g ′ and ( g ′ / g 2 ) -expansion techniques have been customized from the ( w / g ) - expansion method. We applied these two techniques to the paraxial NLS equation and found the optical soliton solutions. The optical soliton solutions are attained as the flat kink, kink, singular kink, peakon, anti-parabolic, W-shape, M-shape, bell, and periodic wave solitons in terms of free parameters. We have presented three-dimensional (3D), two-dimensional (2D) and contour plots of the obtained results and discussed the effect of the free parameters and nonlinearity of the equation by determining different parametric values, which have not been discussed in the previous literature. We have studied the impact of the Kerr nonlinearity and wavenumber on the travelling wave solutions. Moreover, we also analyze the streamlines pattern and instantaneous local directions of the wave profile. All wave phenomena are applied to signal transmission, magneto-acoustic waves in plasma, optical fiber art, coastal engineering, quantum mechanics, hydro-magnetic waves, nonlinear optics and so on. The achieved solutions prove that the proposed methods are very powerful and effective for modern science and engineering for scrutinizing nonlinear evolutionary equations.

Optical Solitons and Modulation Instability Analysis with Lakshmanan–Porsezian–Daniel Model Having Parabolic Law of Self-Phase Modulation

This paper seeks to find optical soliton solutions for Lakshmanan–Porsezian–Daniel (LPD) model with the parabolic law of nonlinearity. The spatiotemporal dispersion is included to the model, as it can contribute to handling the problem of internet bottleneck. This study was performed analytically using the traveling wave hypothesis to reduce the model to an integrable form. Then, the resulting equation was handled with two approaches, namely, the auxiliary equation method and the Bernoulli subordinary differential equation (sub-ODE) method. With an intentional focus on hyperbolic function solutions, abundant optical soliton waves including W-shaped, bright, dark, kink-dark, singular, kink, and antikink solitons were derived with the existing conditions. Furthermore, the behaviors of some optical solitons are illustrated. The spatiotemporal dispersion was found to significantly affect the pulse propagation dynamics. Finally, the modulation instability (MI) of the LPD model is explained in detail along with the extraction of the expression of MI gain.

A governing equation of Rossby waves and its dynamics evolution by Bilinear neural network method

This paper studies an extended evolution equation of large-scale waves by means of bilinear neural network method, which is obtained from local Cartesian coordinate system of the basic equation set by using scaling analysis method and perturbation expansions method. First, we convert the equation into a Hirota equation by using variable transformation. Then, we give the structure framework and the model of the bilinear neural network. We build the test function in two dimensions: depth and breadth. With choosing appropriate activation functions and neuron coefficients, we get many rational function exact solutions, including rogue waves and interaction phenomenon consisting of rogue wave and soliton wave. At last, the figures of these exact solutions are exhibited by selecting suitable value of parameters. We think that these results are very important in ocean dynamics.