Three-dimensional Solitary Waves Research Articles

Three-dimensional nonlinear ion-acoustic solitary waves in the Venusian ionosphere at high altitude

Three-dimensional nonlinear ion-acoustic solitary waves in the Venusian ionosphere at high altitude

Fission of quasi-static dissipative solitons in chiral nematics

Dissipative solitons in liquid crystals (LCs) are represented by three-dimensional solitary waves of director deformation called directrons. The only one exception on the quasi-static counterparts of directrons has ever been observed in achiral nematics. In this work, quasi-static solitons and their fission are identified in chiral nematics. The structure, distribution and fission of quasi-static solitons are closely related to the pitch of samples. The critical pitch is about 7.0 µm for LC cells with thickness of ∼10.0 µm. Quasi-static solitons are transformed from directrons by stepping down voltage to facilitate locating solitons. Successive two-soliton fission with increasing fission time occurs for all quasi-static solitons in samples of relatively larger pitches. Multi-soliton fission is also found in some quasi-static solitons when the voltage is stepped up back to the directron domain, leaving behind a region that can modify the trajectories of surrounding directrons. The fission of quasi-static solitons in chiral nematics has predictable fission location, adjustable fission time, and controllable fission number, may acting as an excellent model system for studying general principles of soliton fission in nonlinear systems.

Low-order modelling of three-dimensional surface waves in liquid film flow on a rotating disk

Using low-dimensional numerical simulations, we investigate the characteristics of complex and three-dimensional surface waves in a liquid film flowing over a rotating disk, focusing on large flow rates from a nozzle. Existing integral boundary layer (IBL) models, which are based on spatially averaged variables along the direction normal to the disk surface, have primarily focused on the formation of axisymmetric waves under relatively small flow rates. In this study, an extended IBL model that accounts for both laminar and turbulent regimes is developed by considering the non-uniformity of the local flow rate in the spreading film flow and incorporating closure models dependent on the local Reynolds number. Our numerical results successfully capture the generation of concentric waves by an impinging circular liquid jet and their transition into three-dimensional solitary waves. These findings are in good agreement with visualization images and time-series data of free-surface fluctuations from a displacement sensor. The backscattering of small-scale three-dimensional turbulence into large-scale horizontal turbulence inside the film plays a critical role in determining the transition of wave modes and the nonlinear dynamics of the waves in the turbulent regime. Furthermore, the behaviour of three-dimensional waves in the downstream region, including frequent wave coalescence in the transition region and the breakup of small-scale solitons, is distinct from that of gravity-driven falling film flows. The amplitude of the three-dimensional waves is inversely related to the generalized Reynolds number defined for rotating films.

Fully Localised Three-Dimensional Gravity-Capillary Solitary Waves on Water of Infinite Depth

Fully localised three-dimensional solitary waves are steady water waves which are evanescent in every horizontal direction. Existence theories for fully localised three-dimensional solitary waves on water of finite depth have recently been published, and in this paper we establish their existence on deep water. The governing equations are reduced to a perturbation of the two-dimensional nonlinear Schrödinger equation, which admits a family of localised solutions. Two of these solutions are symmetric in both horizontal directions and an application of a suitable variant of the implicit-function theorem shows that they persist under perturbations.

Harnessing a multi-dimensional fibre laser using genetic wavefront shaping

The multi-dimensional laser is a fascinating platform not only for the discovery and understanding of new higher-dimensional coherent lightwaves but also for the frontier study of the complex three-dimensional (3D) nonlinear dynamics and solitary waves widely involved in physics, chemistry, biology and materials science. Systemically controlling coherent lightwave oscillation in multi-dimensional lasers, however, is challenging and has largely been unexplored; yet, it is crucial for both designing 3D coherent light fields and unveiling any underlying nonlinear complexities. Here, for the first time, we genetically harness a multi-dimensional fibre laser using intracavity wavefront shaping technology such that versatile lasing characteristics can be manipulated. We demonstrate that the output power, mode profile, optical spectrum and mode-locking operation can be genetically optimized by appropriately designing the objective function of the genetic algorithm. It is anticipated that this genetic and systematic intracavity control technology for multi-dimensional lasers will be an important step for obtaining high-performance 3D lasing and presents many possibilities for exploring multi-dimensional nonlinear dynamics and solitary waves that may enable new applications.

Three-dimensional solitary waves with electrically tunable direction of propagation in nematics

Production of stable multidimensional solitary waves is a grand challenge in modern science. Steering their propagation is an even harder problem. Here we demonstrate three-dimensional solitary waves in a nematic, trajectories of which can be steered by the electric field in a plane perpendicular to the field. The steering does not modify the properties of the background that remains uniform. These localized waves, called director bullets, are topologically unprotected multidimensional solitons of (3 + 2)D type that show fore-aft and right-left asymmetry with respect to the background molecular director; the symmetry is controlled by the field. Besides adding a whole dimension to the propagation direction and enabling controlled steering, the solitons can lead to applications such as targeted delivery of information and micro-cargo.

Long-time behavior of three-dimensional gravity-capillary solitary waves on deep water generated by a moving air-blowing forcing: Numerical study

Long-time behavior of three-dimensional gravity-capillary solitary waves on deep water generated by a moving air-blowing forcing: Numerical study

Stability analysis and applications of traveling wave solutions of three-dimensional nonlinear modified Zakharov–Kuznetsov equation in a magnetized plasma

Propagation of three-dimensional nonlinear solitary waves in a magnetized electron–positron plasma is analyzed. Modified extended mapping method is further modified and applied to three-dimensional nonlinear modified Zakharov–Kuznetsov equation to find traveling solitary wave solutions. As a result, electrostatic field potential, electric field, magnetic field and quantum statistical pressure are obtained with the aid of Mathematica. The new exact solitary wave solutions are obtained in different forms such as periodic, kink and anti-kink, dark soliton, bright soliton, bright and dark solitary waves, etc. The results are expressed in the forms of trigonometric, hyperbolic, rational and exponential functions. The electrostatic field potential and electric and magnetic fields are shown graphically. The soliton stability of these solitary wave solutions is analyzed. These results demonstrate the efficiency and precision of the method that can be applied to many other mathematical physical problems.

Electrically driven three-dimensional solitary waves as director bullets in nematic liquid crystals

Electric field-induced collective reorientation of nematic molecules is of importance for fundamental science and practical applications. This reorientation is either homogeneous over the area of electrodes, as in displays, or periodically modulated, as in electroconvection. The question is whether spatially localized three-dimensional solitary waves of molecular reorientation could be created. Here we demonstrate that the electric field can produce particle-like propagating solitary waves representing self-trapped “bullets” of oscillating molecular director. These director bullets lack fore-aft symmetry and move with very high speed perpendicularly to the electric field and to the initial alignment direction. The bullets are true solitons that preserve spatially confined shapes and survive collisions. The solitons are topologically equivalent to the uniform state and have no static analogs, thus exhibiting a particle–wave duality. Their shape, speed, and interactions depend strongly on the material parameters, which opens the door for a broad range of future studies.

Stability of gravity-capillary solitary waves on shallow water based on the fifth-order Kadomtsev-Petviashvili equation

Longitudinal and transverse instabilities of gravity-capillary solitary waves on shallow water are investigated based on the numerical analysis of the fifth-order Kadomtsev-Petviashvili (KP) equation, which describes the wave phenomena on shallow water where the relevant Bond number is less than and close to 1/3. Two-dimensional (2D) depression gravity-capillary solitary waves are stable to longitudinal perturbations. 2D elevation gravity-capillary solitary waves are unstable to longitudinal perturbations and finally evolve into 2D depression gravity-capillary solitary waves. Three-dimensional (3D) finite-amplitude depression gravity-capillary solitary waves are stable to longitudinal perturbations. 3D finite-amplitude elevation gravity-capillary solitary waves are unstable to longitudinal perturbations and finally evolve into an oscillatory state between two different 3D finite-amplitude depression gravity-capillary solitary waves. 3D small-amplitude depression and elevation gravity-capillary solitary waves are unstable to dilation-type longitudinal perturbations and eventually evolve into an oscillatory state between two different 3D finite-amplitude depression gravity-capillary solitary waves. 3D small-amplitude depression and elevation gravity-capillary solitary waves are unstable to contraction-type longitudinal perturbations and eventually become dispersed out toward still water surface. Finally, 2D depression and elevation gravity-capillary solitary waves are unstable to transverse perturbations and eventually evolve into 3D finite-amplitude depression gravity-capillary solitary waves. Therefore, the only stable gravity-capillary solitary waves on shallow water are 3D finite-amplitude depression gravity-capillary solitary waves. In particular, based on the linear stability analysis, a theoretical proof is presented for the long-wave transverse instability of 2D depression and elevation gravity-capillary solitary waves on shallow water.

Modified KdV–Zakharov–Kuznetsov dynamical equation in a homogeneous magnetised electron–positron–ion plasma and its dispersive solitary wave solutions

Propagation of three-dimensional nonlinear ion-acoustic solitary waves and shocks in a homogeneous magnetised electron–positron–ion plasma is analysed. Modified extended mapping method is introduced to find ion-acoustic solitary wave solutions of the three-dimensional modified Korteweg–de Vries–Zakharov–Kuznetsov equation. As a result, solitary wave solutions (which represent electrostatic field potential), electric fields, magnetic fields and quantum statistical pressures are obtained with the aid of Mathematica. These new exact solitary wave solutions are obtained in different forms such as periodic, kink and antikink, dark soliton, bright soliton, bright and dark solitary wave etc. The results are expressed in the forms of hyperbolic, trigonometric, exponential and rational functions. The electrostatic field potential and electric and magnetic fields are shown graphically. These results demonstrate the efficiency and precision of the method that can be applied to many other mathematical and physical problems.

Vortex solitons in Bose–Einstein condensates with inhomogeneous attractive nonlinearities and a trapping potential

We demonstrate three-dimensional (3D) vortex solitary waves in the (3+1)D nonlinear Gross–Pitaevskii equation (GPE) with spatially modulated nonlinearity and a trapping potential. The analysis is carried out in spherical coordinates, providing for novel localized solutions, and the 3D vortex solitary waves are built that depend on three quantum numbers. Our analytical findings are corroborated by a direct numerical integration of the original equations. It is demonstrated that the vortex solitons found are stable for the quantum numbers n≤2, l≤2 and m=0,1, independent of the propagation distance.

Vector matter waves in two-component Bose-Einstein condensates with spatially modulated nonlinearities

We demonstrate three-dimensional (3D) vector solitary waves in the coupled (3 + 1)-D nonlinear Gross-Pitaevskii equations with variable nonlinearity coefficients. The analysis is carried out in spherical coordinates, providing novel localized solutions that depend on three modal numbers, l, m, and n. Using the similarity transformation (ST) method in 3D, vector solitary waves are built with the help of a combination of harmonic and trapping potentials, including multipole solutions and necklace rings. In general, the solutions found are stable for low values of the modal numbers; for values larger than 2, the solutions are found to be unstable. Variable nonlinearity allows the utilization of soliton management methods.

A new ZK–BO equation for three-dimensional algebraic Rossby solitary waves and its solution as well as fission property

The BO equation depicts the evolution of algebraic Rossby solitary waves in ocean and atmosphere. But, despite its overt fame, the BO equation is restricted as a two-dimensional model. On the basis of the great success in the soliton theory, a lot of works have recently been directed to three-dimensional models and investigations of solitary waves properties in three-dimensional systems. In this paper, a new ZK–BO equation for three-dimensional algebraic Rossby solitary waves is derived by employing perturbation expansions and stretching transformations of time and space. By virtue of the trial function method, the exact solution of the ZK–BO equation is presented. By comparing the solution with the solution of two-dimensional algebraic Rossby solitary waves, we can find that the wave length of three-dimensional algebraic Rossby solitary waves is shorter while the wave amplitude is higher. Based on the exact solution of ZK–BO equation, the dissipation effect is studied. The results reveal the influence of dissipation effect on the three-dimensional algebraic solitary waves. Further, after theoretical analysis, the conservation laws of three-dimensional algebraic Rossby solitary waves are discussed and the fission property is studied.

Modeling of multidimensional light bullets in Fermi liquid and ADS/CFT correspondence

This paper discusses two-dimensional and three-dimensional solitary electromagnetic waves propagation in Fermi liquid in the framework of ADS/CFT correspondence. The electromagnetic field is classically considered within the framework of Maxwell's equations. The obtained effective equation is numerically analyzed and the state of the electromagnetic field that is localized in two / three spatial dimensions is revealed.

Amplitude modulation of three-dimensional low-frequency solitary waves in a magnetized dusty superthermal plasma

The amplitude modulation of three-dimensional (3D) dust ion-acoustic wave (DIAW) packets is studied in a collisionless magnetized plasma with inertial positive ions, superthermal electrons and negatively charged immobile dust grains. By using the reductive perturbation technique, a 3D-nonlinear Schrödinger equation is derived, which governs the slow modulation of DIAW packets. The latter are found to be stable in the low-frequency (omega <omega _{text {c}}) regime, whereas they are unstable for omega >omega _{text {c}}, and the modulational instability is related to the modulational obliqueness (theta ). Here, omega ~(omega _{text {c}}) is the nondimensional wave (ion-cyclotron) frequency. It is shown that the superthermal parameter kappa, the frequency omega _{text {c}} as well as the charged dust impurity (0<mu <1) shift the MI domains around the omega–theta plane, where mu is the ratio of electron-to-ion number densities. Furthermore, it is found that the decay rate of instability is quenched by the superthermal parameter kappa with cutoffs at lower wave number of modulation (K); however, it can be higher (lower) with increasing values of mu (omega _{text {c}}) having cutoffs at higher values of K.

Light bullets in coupled nonlinear Schrödinger equations with variable coefficients and a trapping potential.

We analyze three-dimensional (3D) vector solitary waves in a system of coupled nonlinear Schrödinger equations with spatially modulated diffraction and nonlinearity, under action of a composite self-consistent trapping potential. Exact vector solitary waves, or light bullets (LBs), are found using the self-similarity method. The stability of vortex 3D LB pairs is examined by direct numerical simulations; the results show that only low-order vortex soliton pairs with the mode parameter values n ≤ 1, l ≤ 1 and m = 0 can be supported by the spatially modulated interaction in the composite trap. Higher-order LBs are found unstable over prolonged distances.

Observations of gravity–capillary lump interactions

In this experimental study, we investigate the interaction of gravity–capillary solitary waves generated by two surface pressure sources moving side by side at constant speed. The nonlinear response of a water surface to a single source moving at a speed just below the minimum phase speed of linear gravity–capillary waves in deep water ($c_{min}\approx 23~\text{cm}~\text{s}^{-1}$) consists of periodic generation of pairs of three-dimensional solitary waves (or lumps) in a V-shaped pattern downstream of the source. In the reference frame of the laboratory, these unsteady lumps propagate in a direction oblique to the motion of the source. In the present experiments, the strengths of the two sources are adjusted to produce nearly identical responses and the free-surface deformations are visualized using photography-based techniques. The first lumps generated by the two sources move in intersecting directions that make a half-angle of approximately $15^{\circ }$ and collide in the centreplane between the sources. A steep depression is formed during the collision, but this depression quickly decreases in amplitude while radiating small-amplitude radial waves. After the collision, a quasi-stable pattern is formed with several rows of localized depressions that are qualitatively similar to lumps but exhibit periodic amplitude oscillations, similar to a breather. The shape of the wave pattern and the period of oscillations depend strongly on the distance between the sources.

Comparison of Analytical and Numerical Simulations of Long Nonlinear Internal Solitary Waves in Shallow Water

ABSTRACT Mohapatra, S.C.; Fonseca, R.B., and Guedes Soares, C., 2018. Comparison of analytical and numerical simulations of long nonlinear internal solitary waves in shallow water. This study compares the results of long nonlinear internal waves between analytical and numerical wave models in shallow water. To present the analytical solutions, the (G′/G) expansion method is applied to obtain the exact and explicit nonlinear solitary wave solutions of the generalized Gardner equation of any order in the presence of dispersive coefficient. The internal waves are demonstrated by analysing the solutions of the classical Gardner equation, modified Korteweg-de Vries equation, and Korteweg-de Vries equations from the generalized solution. The shape and characteristics of the three-dimensional solitary waves are studied by analysing different numerical results of the nonlinear analytical solutions. The Gardner equation basically behaves with the same properties as the Boussinesq equations of travelling wave solut...

ZK-Burgers equation for three-dimensional Rossby solitary waves and its solutions as well as chirp effect

Two-dimensional Rossby solitary waves propagating in a line have attracted much attention in the past decade, whereas there is few research on three-dimensional Rossby solitary waves. But as is well known, three-dimensional Rossby solitary waves are more suitable for real ocean and atmosphere conditions. In this paper, using multiscale and perturbation expansion method, a new Zakharov-Kuznetsov (ZK)-Burgers equation is derived to describe three-dimensional Rossby solitary waves that propagate in a plane. By analyzing the equation we obtain the conservation laws of three-dimensional Rossby solitary waves. Based on the sine-cosine method, we give the classical solitary wave solutions of the ZK equation; on the other hand, by the Hirota method we also obtain the rational solutions, which are similar to the solutions of the Benjamin-Ono (BO) equation, the solutions of which can describe the algebraic solitary waves. The rational solutions of the ZK equations are worth of attention. Finally, with the help of the classical solitary wave solutions, similar to the fiber soliton communication, we discuss the dissipation and chirp effect of three-dimensional Rossby solitary waves.