Types Of Solitary Waves Research Articles

Dynamical analysis of rational and semi‐rational solution for a new extended (3 + 1)‐dimensional Kadomtsev‐Petviashvili equation

This paper proposes a new extended (3 + 1)‐dimensional Kadomtsev‐Petviashvili equation that portrays a unique dispersion effect about . Its integrability is confirmed via the WTC‐Kruskal algorithm in Painlevé sense. ‐soliton, breather, and ‐type solitary wave are derived systematically at first. Then, the mixed solution composed of soliton and breather is obtained. In addition, the “long wave” limit is employed to construct rational and semi‐rational solution. The rational solution can be classified as rogue wave, ‐type solitary wave, and lump wave. The semi‐rational solution has the form a hybrid of two solitons, a hybrid of rogue wave and soliton, a hybrid of lump and soliton(s), and a hybrid of lump and breather. The results may help simulate complex waves and their interactions in fluid.

Existence of Davey--Stewartson Type Solitary Waves for the Fully Dispersive Kadomtsev--Petviashvilii equation

We prove existence of small-amplitude modulated solitary waves for the full-dispersion Kadomtsev--Petviashvilii (FDKP) equation with weak surface tension. The resulting waves are small-order perturbations of scaled, translated and frequency-shifted solutions of a Davey--Stewartson (DS) type equation. The construction is variational and relies upon a series of reductive steps which transform the FDKP functional to a perturbed scaling of the DS functional, for which least-energy ground states are found. We also establish a convergence result showing that scalings of FDKP solitary waves converge to ground states of the DS functional as the scaling parameter tends to zero. Our method is robust and applies to nonlinear dispersive equations with the properties that (i) their dispersion relation has a global minimum (or maximum) at a non-zero wave number, and (ii) the associated formal weakly nonlinear analysis leads to a DS equation of elliptic-elliptic focussing type. We present full details for the FDKP equation.

Simulation of hyperbolic type solitary waves based on velocity parameter for (3 + 1)-dimensional the B-type Kadomtsev–Petviashvili–Boussinesq equation

In this paper, we investigate solutions to the [Formula: see text]-dimensional B-type Kadomtsev–Petviashvili–Boussinesq (B-KPB) equation, which has strong nonlinearity feature by using sub-equation and the modified [Formula: see text]-expansion methods. Using these analytical methods, dark soliton, trigonometric oscillation, singular soliton, algebraic and hyperbolic type oscillating traveling wave solutions are presented. By giving special values to the parameters in the obtained traveling wave solutions, 3D, 2D and contour graphs representing the solitary waves are drawn. The advantages and disadvantages of both analytical methods are presented in comparison with each other. In addition, the changes of the solutions of the B-KPB equation according to the velocity parameter and the refraction phenomenon are supported by simulation. In addition, the solutions of the equation discussed are first discussed in detail, physical interpretations. Secondly, the geometric interpretations of the traveling wave solutions are analyzed and it is shown that the graphs of the solutions correspond to flat hyper-surfaces. It is argued that these hyper-surfaces can be leveled on a hyperplane without distortion.

Abundant variant wave patterns by coupled Boussinesq–Whitham–Broer–Kaup equations

This paper focus on searching for abundant wave patterns to time-fractional coupled Boussinesq–Whitham–Broer–Kaup equations, which plays a crucial role in obtaining the novel insights for propagational dynamics of shallow water. Through the innovative application of complete discrimination system for polynomial method (CDSPM) and conformable fractional transformation, the whole types of solutions for the fractional equations presented in the existing literatures are obtained. In particular, we obtain the solitary wave type solutions and Jacobian elliptic functions type solutions, and the later are initially founded. The existence and applicability of the obtained solutions are proved by concrete examples. Finally, the nature of wave propagation and the fractional characteristics are intuitively revealed by graphical simulations.

Interfacial electrohydrodynamic solitary waves under horizontal electric fields

Interfacial waves between two superimposed dielectric fluid layers under a horizontal electric field are investigated from asymptotic and numerical aspects. The fluid is taken to be inviscid, incompressible and non-conducting in each layer. The competing forces resulting from gravity, surface tension and electric field are all considered. A systematic procedure is proposed to derive model equations of multiple scales in various possible limits from the electrified Euler equations in the framework of the Zakharov–Craig–Sulem formulation. Based on thorough analyses of the Dirichlet–Neumann operators in the long-wave approximation, classic weakly nonlinear models – including the Boussinesq-type system, the Korteweg–de Vries (KdV) equation and its variants, and the Benjamin-type equation – are obtained under different scaling assumptions. In addition, strongly nonlinear models (without the smallness assumption on the wave amplitude) in the Benjamin and Barannyk–Papageorgiou–Petropoulos regimes are derived. In these models, the electric effects are shown to produce dispersive regularisations of long and short waves. A modified boundary integral equation method is developed to compute solitary waves in the original electrified Euler equations. Through comparisons with solitary-wave solutions in the Euler equations, it is found that in various asymptotic regimes, weakly nonlinear models are in overall good agreement when wave amplitudes are small. In contrast, the range of validity of the strongly nonlinear model is much broader. It is shown that the horizontal electric field plays a significant role in the physical system: it expands the range of parameters for the existence of progressive waves, changes the qualitative characteristics of solitary waves and leads to a new type of solitary wave, namely the KdV–wavepacket mixed type.

Spatiotemporal nonlinear evolution of the laser pulse and turbulence generation in laser produced plasmas

This study presents a model to understand the behavior of the turbulence generated in the magnetic field of mega gauss order during high-intensity laser interaction with magnetized plasma. The modified nonlinear Schrödinger (MNLS) equation is developed by contemplating the effect of the group velocity dispersion, diffraction, and nonlinearity induced by the relativistic variation of electron mass and the nonlinear ponderomotive force. Numerical simulation is carried out to solve the dimensionless MNLS equation. The simulation results show the generation of the solitary wave type coherent structures in the nonlinear spatiotemporal evolution of the laser pulse at the early stage, but subsequent turbulence generation has also been observed. The ensemble-averaged turbulent power spectrum has been studied and the power-law scaling is approximately ∼ k−1.85(a solid red line of scaling k−1.85 is given for reference). To get insight into the spatiotemporal nonlinear development of the laser pulse, while propagating in the plasma medium, a semi-analytical model has also been presented. The present study could be substantial in replicating astrophysical scenarios by laboratory simulations along with understanding the underlying quintessential physics of magnetic turbulence.

Hydroelastic lumps in shallow water

Hydroelastic solitary waves propagating on the surface of a three-dimensional ideal fluid through the deformation of an elastic sheet are studied. The problem is investigated based on a Benney–Luke-type equation derived via an explicit non-local formulation of the classic water wave problem. The normal form analysis is carried out for the newly developed equation, which results in the Benney–Roskes–Davey–Stewartson (BRDS) system governing the coupled evolution of the envelope of a carrier wave and the wave-induced mean flow. Numerical results show three types of free solitary waves in the Benney–Luke-type equation all of which are predicted by the BRDS system: plane solitary wave, lump (i.e., fully localized traveling waves in three dimensions), and transversally periodic solitary wave. They are linked together by a dimension-breaking bifurcation where plane solitary waves and lumps can be viewed as two limiting cases, and transversally periodic solitary waves serve as intermediate states. The stability and interaction of solitary waves are investigated via a numerical time integration of the Benney–Luke-type equation. For a localized load moving on the elastic sheet with a constant speed, it is found that there exists a transcritical regime of forcing speed for which there are no steady solutions. Instead, periodic shedding of lumps can be observed if the forcing moves at speed in this range.

An experimental study on the motion of solid spheres under solitary wave attack

In the present study, the motion of a single and the motion and collision of two spheres on a horizontal area under solitary wave attack are investigated, constituting a unique and integrated dataset. Sixteen experimental cases are considered on four experimental configurations changing the number of spheres, the porosity of the bottom, water levels, and the breaking condition of the solitary waves (non-breaking or breaking). The motion of the spheres is tracked using a color-detection-based image-processing algorithm. The experiments reveal four significant results: (i) The rate of damping on the porous bottom is 1.65 times of impermeable bottom on average. (ii) The water level changes the characteristics of the motion of the spheres significantly due to the friction forces and vortex structures created by the interaction of waves with moving spheres. (iii) The dimensionless terminal velocities of the spheres in the non-breaking cases are 1.38 times of breaking solitary wave cases on average. (iv) The type of solitary wave and the water level significantly affect the collision behavior; however, the bottom condition does not have an apparent contribution to the collision behavior.

Resonant collisions among two-dimensional localized waves in the Mel’nikov equation

We study the resonant collisions among different types of localized solitary waves in the Mel’nikov equation, which are described by exact solutions constructed using Hirota direct method. The elastic collisions among different solitary waves can be transformed into resonant collisions when the phase shifts of these solitary waves tend to infinity. First, we study the resonant collision among a breather and a dark line soliton. We obtain two collision scenarios: (i) the breather is semi-localized in space and is not localized in time when it obliquely intersects with the dark line soliton, and (ii) the breather is semi-localized in time and is not localized in space when it parallelly intersects with the dark line soliton. The resonant collision of a lump and a dark line soliton, as the limit case of resonant collision of a breather and a dark line soliton, shows the fusing process of the lump into the dark line soliton. Then, we investigate the resonant collision among a breather and two dark line solitons. In this evolution process, we also obtain two dynamical behaviors: (iii) when the breather and the two dark line solitons obliquely intersect each other, we get that the breather is completely localized in space and is not localized in time, and (iv) when the breather and the two dark line solitons are parallel to each other, we get that the breather is completely localized in time and is not localized in space. The resonant collision of a lump and two dark line solitons is obtained as the limit case of the resonant collision among a breather and two dark line solitons. In this special case, the lump first detaches from a dark line soliton and then disappears into the other dark line soliton. Eventually, we also investigate the intriguing phenomenon that when a resonant collision among a breather and four dark line solitons occurs, we get the interesting situation that two of the four dark line solitons are degenerate and the corresponding solution displays the same shape as that of the resonant collision among a breather and two dark line solitons, except for the phase shifts of the solitons, which are not only dependent of the parameters controlling the waveforms of the solitons and the breather, but also dependent of some parameters irrelevant to the waveforms.

Solitary matter wave in spin-orbit-coupled Bose-Einstein condensates with helicoidal gauge potential.

We analytically and numerically study the different types of solitary wave in the two-component helicoidal spin-orbit coupled Bose-Einstein condensates (BECs). Adopting the multiscale perturbation method, we derive the analytical bright and dark solitary wave solutions of the system, and the stationary and moving bright (dark) solitary waves are obtained. The effects of spin-orbit coupling, the helicoidal gauge potential, the momentum, the Zeeman splitting, and the atomic interactions on the solitary wave types are discussed, and it is found that the coupling of these physical parameters can manipulate different types of solitary waves in the system. The results indicate that the helicoidal gauge potential breaks the symmetric properties of the energy band of the system and adjusts the energy band structure, thus further effecting the solitary wave properties, i.e., stationary or moving solitary wave, bright, or dark solitary wave. Correspondingly, the analytical predictions for exciting stationary or moving bright (dark) solitary wave in parameter space are obtained. In particular, the helicoidal gauge potential changes the solitary wave types drastically for the weak spin-orbit coupling, i.e., in the absence of the helicoidal gauge potential, only dark (bright) solitary wave solutions exist in the system with repulsive (attractive) atomic interaction; however, in the presence of the helicoidal gauge potential, both dark and bright solitary waves can exist in the system regardless of whether the atomic interaction is repulsive or attractive. In addition, we investigate the stability of solitary waves and obtain the stability regions of different types of solitary waves by applying the linear stability analysis. The dynamic evolution results of the solitary waves by the direct numerical simulation not only validate the linear stability analysis but also confirm the analytical prediction of the solitary waves. Finally, the collision effects between solitary waves are also presented by the numerical simulation. It is shown that the interactions between solitary waves in the system have both elastic and inelastic collisions, which are closely related to the position of solitary wave states in the linear energy band. Our results provide a potential way to adjust the types of solitary waves in BECs with helicoidal gauge potential.

Simulation and refraction event of complex hyperbolic type solitary wave in plasma and optical fiber for the perturbed Chen-Lee-Liu equation

In this article, Modified \(1/G^{\prime}\)-expansion and Modified Kudryashov methods are applied to generate traveling wave solutions of Perturbed Chen-Lee-Liu equation. The similar and different aspects of the solutions produced by both analytical methods are discussed. By giving special values to the constants in the solutions obtained by analytical methods, 2D, 3D and contour graphics representing the shape of the standing wave at any time are presented. Additionally, the advantages and disadvantages of the two analytical methods are discussed and presented. Also, a solitary wave is produced by giving special values to the parameters in the hyperbolic type complex traveling wave solution. Simulations are created for different values of the amplitute and velocity propagation parameters of the solitary wave. The values of these parameters are calculated for the breakage event physically. A computer package program is used for operations such as solving complex operations, drawing graphics and solving systems of algebraic equations.

Interaction of Schamel type solitary waves in an electronegative plasma with trapped non-extensive electrons

Interaction of Schamel type solitary waves in an electronegative plasma with trapped non-extensive electrons

Wave propagation with degenerate electron–positron in magnetically quantised ion beam plasma

In an ion beam plasma system, the effect of magnetically quantised degenerate trapped electron and positron, on small-amplitude ion-acoustic solitary waves are studied with the help of Korteweg–de Vries (KdV) equation. Here the magnetised positive ions and beam ions are considered as non-degenerate. The effect of magnetic quantisation, degenerate temperature, normalised positron concentration, normalised ion beam concentration along with other relevant physical plasma parameters of the origin of astrophysical plasma environment, on solitary wave propagation is studied especially in the nonlinear regime. On the other hand, in the linear regime, the dependency of frequency and wavenumber on the aforementioned plasma parameters are discussed. It is found that both compressive and rarefactive type solitary waves can exist in such a plasma environment. Three different modes of propagation, the fast beam and slow beam modes and the inherent ion-acoustic mode coexist in such plasma systems and the role of different degenerate plasma parameters on these wave modes are also discussed in detail. The normalised positron concentration and velocity, as well as the normalised ion beam concentration along with different degenerate plasma parameters, has an astounding control on the amount of compression and rarefaction, a typical plasma wave mode is supposed to undergo. Within the chosen degenerate and non-degenerate plasma parameters, for a compressive (rarefactive) fast beam mode, the slow mode appears as rarefactive (compressive). For some typical combinations of degenerate parameters, the fast mode can propagate with a hypersonic phase velocity which shows distinctive characteristics with positron density.

Multiple explicit solutions of the 2D variable coefficients Chafee–Infante model via a generalized expansion method

In this work, we do apply a generalized expansion method to the realistic two-dimensional (2D) Chafee–Infante model with time-variable coefficients which is encountered in physical sciences.The key ideas of this method consist in: (i) to choose a nonlinear wave variable in respect to time-variable into the general finite series solution of the governing model; (ii) to take a full advantage from the general elliptic equation introduced as an auxiliary equation which can degenerate into sub-equations such as Riccati equation, the Jacobian elliptic equations, the generalized Riccati equation. Based upon this powerful technique, we successfully construct for the first time several types of non-autonomous solitary waves as well as some non-autonomous triangular solutions, rational or doubly periodic type ones. We investigate the propagation of non-autonomous solitons and we emphasize as well upon the influence of the variable coefficients. We are providing and analyzing a few graphical representations of some specific solutions. The results of this paper will be valuable in the study of nonlinear physical phenomena. The above- mentioned method could be employed to solve other partial differential equations with variable coefficients which describe various complicated natural phenomena.

Periodic, kink and bell shape wave solutions to the Caudrey-Dodd-Gibbon (CDG) equation and the Lax equation

This paper addresses the implementation of the new generalized ((G')⁄G)-expansion method to the Caudrey-Dodd-Gibbon (CDG) equation and the Lax equation which are two special case of the fifth-order KdV (fKdV) equation. The method work well to derive a new variety of travelling wave solutions with distinct physical structures such as soliton, singular soliton, kink, singular kink, bell-shaped soltion, anti-bell-shaped soliton, periodic, exact periodic and bell type solitary wave solutions. Solutions provided by this method are numerous comparing to other methods. To understand the physical aspects and importance of the method, solutions have been graphically simulated. Our results unquestionably disclose that new generalized ((G')⁄G)-expansion method is incredibly influential mathematical tool to work out new solutions of various types of nonlinear partial differential equations arises in the fields of applied sciences and engineering.

Onset of the broad-ranging general stable soliton solutions of nonlinear equations in physics and gas dynamics

Stable soliton solutions for the nonlinear Klein–Gordon equation in condensed matter physics, particle physics, nonlinear optics, solid state physics and the gas dynamics equation ensuing in shock fronts have been established by putting use of the sine-Gordon expansion procedure. We establish the broad-ranging familiar stable wave solutions related with some free parameters, for particular values of which some of the subsisting solutions in the literature are re-established and several fresh solutions are formulated. The contour and 3D shapes are depicted to describe the physical phenomena of the established solutions. Scores of solitary wave solutions are obtained such as, compacton, smooth soliton, anti-bell type solitary wave, bell-type solitary wave, kink and some other new types of solitary wave solutions. The obtained results might play important role in optics, quantum mechanics, mathematical physics and engineering. It is noticeable to inspect that the sine-Gordon expansion procedure is an efficient, competent and straightforward mathematical tool to ascertain exact solitary wave solutions.

Solitary waves dynamic for Davydov α-helical protein model: Effects of localized and periodic inhomogeneities.

We describe the dynamic of excitons through single inhomogeneous α-helical proteins with off-diagonal and diagonal couplings. Inhomogeneities considered are either localized or periodic. Intensive numerical simulations carried show stable structures and allow us to single out important features of excitons dynamic. In the absence of inhomogeneities, the interplay between off-diagonal and diagonal couplings leads to two distinct types of solitary waves. Bright solitary waves correspond to an off-diagonal coupling constant lower than a critical diagonal coupling constant, while dark solitary waves are obtained in the opposite case. Inclusion of inhomogeneities profoundly affects the profiles, amplitudes, and energies transported by the waves. For relatively small strength of inhomogeneities, only the profiles of the waves significantly change, the other properties remaining almost unchanged. Large strength inhomogeneities sensitively twist the profiles and increase amplitudes and energies of the waves. Our study suggests that small strength inhomogeneities allow a coherent transport of energy and the biological functions remain unchanged, but large strengths of inhomogeneities affect the biological functioning of the α-helical protein chains. Hence, large strengths of inhomogeneities may amplify the energy of the molecule and could be used to treat some diseases.

Dynamics and spectral stability of soliton-like structures in fluid-filled membrane tubes

This survey presents results on the stability of elevation solitary waves in axisymmetric elastic membrane tubes filled with a fluid. The elastic tube material is characterized by an elastic potential (elastic energy) that depends non-linearly on the principal deformations and describes the compliant elastic media. Our survey uses a simple model of an inviscid incompressible fluid, which nevertheless makes it possible to trace the main regularities of the dynamics of solitary waves. One of these regularities is the spectral stability (linear stability in form) of these waves. The basic equations of the ‘axisymmetric tube – ideal fluid’ system are formulated, and the equations for the fluid are averaged over the cross-section of the tube, that is, a quasi-one-dimensional flow with waves whose length significantly exceeds the radius of the tube is considered. The spectral stability with respect to axisymmetric perturbations is studied by constructing the Evans function for the system of basic equations linearized around a solitary wave type solution. The Evans function depends only on the spectral parameter , is analytic in the right-hand complex half-plane , and its zeros in coincide with unstable eigenvalues. The problems treated include stability of steady solitary waves in the absence of a fluid inside the tube (the case of constant internal pressure), together with the case of local inhomogeneity (thinning) of the tube wall, the presence of a steady fluid filling the tube (the case of zero mean flow) or a moving fluid (the case of non-zero mean flow), and also the problem of stability of travelling solitary waves propagating along the tube with non-zero speed. Bibliography: 83 titles.

Computing solitary wave solutions of coupled nonlinear Hirota and Helmholtz equations

In this article, we obtain the exact solutions two coupled models, one integrable system, namely coupled nonlinear Hirota (CNHI) equation and another non-integrable system, namely coupled nonlinear Helmholtz (CNHE) equation via the exp (−Φ(ε))–expansion method. The obtained travelling wave solutions are structured in rational, trigonometric and hyperbolic functions. These solutions lead to diverse types of solitary optical waves for free choices of parameters that guarantee the sustainability of such solutions. Also, 3D illustrations for the free choices of the physical parameters is provided to understand the physical explanation of the problems. These results further enrich and deepen the understanding of the dynamics of higher-dimensional soliton propagation.

Analytical approximate solutions for nonplanar Burgers equations by weighted residual method

In this work, analytical approximate progressive wave solutions for the generalized form of the nonplanar KdV-Burgers (KdV-B) and mKdV-Burgers (mKdV-B) equations are presented and the results are discussed. Motivated with the exact solutions of the planar KdV-B and mKdV-B equations, the weighted residual method is applied to propose analytical approximate solutions for the generalized form of the nonplanar KdV-B and mKdV-B equations. The structure of the KdV-B equation assumes a solitary wave type of solution, whereas the mKdV-B equation assumes a shock wave type of solution. The analytical approximate progressive wave solutions for the cylindrical(spherical) KdV-B and mKdV-B equations are obtained as some special cases and compared with numerical solutions and the results are depicted on 2D and 3D figures. The results revealed that both solutions are in good agreement. The advantage of the present method is that it is rather simple as compared to the inverse scattering method and gives the same results with the perturbative inverse scattering technique. Moreover, the present analytical solutions allow readers to carry out physical parametric studies on the behavior of the solution. In addition to the present solutions are defined overall the problem domain not only over the grid points, as well as the solution calculation has less CPU time-consuming and round-off error.