Soliton-like Waves Research Articles

Approaching Electroencephalographic Pathological Spikes in Terms of Solitons

A delicate balance between dissipative and nonlinear forces allows traveling waves termed solitons to preserve their shape and energy for long distances without steepening and flattening out. Solitons are so widespread that they can generate both destructive waves on oceans’ surfaces and noise-free message propagation in silica optic fibers. They are naturally observed or artificially produced in countless physical systems at very different coarse-grained scales, from solar winds to Bose–Einstein condensates. We hypothesize that some of the electric oscillations detectable by scalp electroencephalography (EEG) could be assessed in terms of solitons. A nervous spike must fulfill strict mathematical and physical requirements to be termed a soliton. They include the proper physical parameters like wave height, horizontal distance and unchanging shape; the appropriate nonlinear wave equations’ solutions and the correct superposition between sinusoidal and non-sinusoidal waves. After a thorough analytical comparison with the EEG traces available in the literature, we argue that solitons bear striking similarities with the electric activity recorded from medical conditions like epilepsies and encephalopathies. Emerging from the noisy background of the normal electric activity, high-amplitude, low-frequency EEG soliton-like pathological waves with relatively uniform morphology and duration can be observed, characterized by repeated, stereotyped patterns propagating on the hemispheric surface of the brain over relatively large distances. Apart from the implications for the study of cognitive activities in the healthy brain, the theoretical possibility to treat pathological brain oscillations in terms of solitons has powerful operational implications, suggesting new therapeutical options to counteract their detrimental effects.

Streaming instability in neutron star magnetospheres: No indication of soliton-like waves

Context. Coherent radiation of pulsars, magnetars, and fast radio bursts could, in theory, be interpreted as radiation from solitons and soliton-like waves. Solitons are meant to contain a large number of electric charges confined on long timescales and can radiate strongly via coherent curvature emission. However, solitons are also known to undergo a wave collapse, which casts doubts on the correctness of the soliton radio emission models of neutron stars. Aims. We investigated the evolution of the caviton type of solitons self-consistently formed by the relativistic streaming instability and compared their apparent stability in 1D calculations with more generic 2D cases, in which the solitons are seen to collapse. Three representative cases of beam Lorentz factors and plasma temperatures were studied to obtain soliton dispersion properties. Methods. We utilized 1D electrostatic and 2D electromagnetic relativistic particle-in-cell simulations at kinetic microscales. Results. We find that no solitons are generated by the streaming instability in the 2D simulations. Only superluminal L-mode (relativistic Langmuir) waves are produced during the saturation of the instability, but these waves have smaller amplitudes than the waves in the 1D simulations. The amplitudes tend to decrease after the instability has saturated, and only waves close to the light line, ω = ck, remain. Solitons in the 1D approach are stable for γb ≳ 60, but they disappear for low beam Lorentz factors, γb < 6. Conclusions. Our examples show that the superluminal soliton branch that is formed in 1D simulations will not be generated by the relativistic streaming instability when more dimensional degrees of freedom are present. The soliton model cannot, therefore, be used to explain the coherent radiation of pulsars, magnetars, and fast radio bursts – unless one can show that there are alternative plasma mechanisms for the soliton generation.

Nonlinear charge and energy transport in anharmonic quasi-two-dimensional systems.

We study the localized states of an extra electron in an anisotropic quasi-two-dimensional system in which the electron-lattice interaction and the anharmonicity of the lattice vibrations are dominant in one direction. This model describes layers of polydiacetylene or other polymer chains, beta sheets of polypeptides, multilevel microstructures of conjugated polymers, and other low-dimensional systems. It is shown that for appropriate parameter values of the system an extra electron can excite a soliton-like mobile wave of the lattice deformation, within which it can get self-trapped. Such a bound state of an electron and the lattice deformation form a nonlinear two-component polaron-like entity, which can propagate with minimum of the energy dissipation. Our findings are based on the variational approach and the full numerical solution of the coupled system of nonlinear equations. These results suggest that the experimentally measured charge and energy transport over macroscopic distances in the above-mentioned systems can be provided by the soliton mechanism and thus have a potential impact on the theoretical background of the numerous applications of low-dimensional materials in nanoelectronics.

Dirac solitons and topological edge states in the β-Fermi-Pasta-Ulam-Tsingou dimer lattice.

We consider a dimer lattice of the Fermi-Pasta-Ulam-Tsingou (FPUT) type, where alternating linear couplings have a controllably small difference and the cubic nonlinearity (β-FPUT) is the same for all interaction pairs. We use a weakly nonlinear formal reduction within the lattice band gap to obtain a continuum, nonlinear Dirac-type system. We derive the Dirac soliton profiles and the model's conservation laws analytically. We then examine the cases of the semi-infinite and the finite domains and illustrate how the soliton solutions of the bulk problem can be glued to the boundaries for different types of boundary conditions. We thus explain the existence of various kinds of nonlinear edge states in the system, of which only one leads to the standard topological edge states observed in the linear limit. We finally examine the stability of bulk and edge states and verify them through direct numerical simulations, in which we observe a solitonlike wave setting into motion due to the instability.

The exotic behavior of the wave evolution in Lévy crystals within a fractional medium

We investigate a traveling Gaussian wave packet transport through a rectangular quantum barrier of lévy crystals in fractional quantum mechanics formalism. We study both standard and fractional Schrödinger equations in linear and nonlinear regimes by using a split-step finite difference (SSFD) method. We evaluate the reflection, trapping, and transmission coefficients of the wave packet and the wave packet spreading by using time-dependent inverse participation ratio (IPR) and second moment. By simultaneously adjusting the fractional and nonlinear terms, we create sharp pulses, which is an essential issue in optoelectronic devices. We illustrate that the effects of barrier height and width on the transmission coefficient are strangely different for the standard and fractional Schrödinger equations. We observe fortunately soliton-like localized wave packets in the fractional regime. Thus, we can effectively control the behavior of the wave evolution by adjusting the available parameters, which can excite new ideas in optics.

Mathematical modeling of chirped modulated waves along a multi-coupled nonlinear electrical transmission line with dispersive elements

This work presents a mathematical modeling of chirped modulated waves propagating along a two-dimensional nonlinear electrical transmission line consisting of nonlinear dispersive and purely nonlinear capacitive elements. Combining the rotating-wave approximation technique with the perturbation method, we derive a two-dimensional extended nonlinear Schrödinger equation with self-steepening and self-frequency shift, that predicts the formation and propagation of a very rich variety of soliton-like waves including bright, dark, kink, and double-kink soliton-like waves with nonlinear chirp in a multi-coupled nonlinear electrical transmission network with dispersive elements. Using this model equation, parameter domains are delineated in which these modulated waves exist. The parameter domains are found to be strongly dependent on the dispersive elements of the network system. Through exact solutions of the amplitude equation, we investigate the dynamics of chirped modulated waves along the network system under consideration. Our results show that the nonlinear chirp associated with each of the found soliton-like pulses is directly proportional to the wave intensity and is saturated at some finite value as the propagation coordinate approaches its asymptotic value. Also, we show that increasing the number of the network lines decreases the propagating speed of the soliton-like pulses. Most importantly, theoretical analysis show that chirping of found network pulses can be controlled by varying the self-steepening term and self-frequency shift. The validity of analytical results and the robustness of these chirped soliton-like voltage signals which may have important applications in signal processing systems are confirmed by numerical simulations. The results of our study may be launched in long distance telecommunication lines involving higher-order nonlinearities of the fiber.

Quaternion Quantum Mechanics II: Resolving the Problems of Gravity and Imaginary Numbers

We present a quaternion representation of quantum mechanics that allows its ontological interpretation. The correspondence between classical and quaternion quantum equations permits one to consider the universe (vacuum) as an ideal elastic solid. Elementary particles would have to be standing or soliton-like waves. Tension induced by the compression and twisting of the elastic medium would increase energy density, and as a result, generate gravity forcing and affect the wave speed. Consequently, gravity could be described by an index of refraction.

Characteristics of New Stochastic Solitonic Solutions for the Chiral Type of Nonlinear Schrödinger Equation

The Wiener process was used to explore the (2 + 1)-dimensional chiral nonlinear Schrödinger equation (CNLSE). This model outlines the energy characteristics of quantum physics’ fractional Hall effect edge states. The sine-Gordon expansion technique (SGET) was implemented to extract stochastic solutions for the CNLSE through multiplicative noise effects. This method accurately described a variety of solitary behaviors, including bright solitons, dark periodic envelopes, solitonic forms, and dissipative and dissipative–soliton-like waves, showing how the solutions changed as the values of the studied system’s physical parameters were changed. The stochastic parameter was shown to affect the damping, growth, and conversion effects on the bright (dark) envelope and shock-forced oscillatory wave energy, amplitudes, and frequencies. In addition, the intensity of noise resulted in enormous periodic envelope stochastic structures and shock-forced oscillatory behaviors. The proposed technique is applicable to various energy equations in the nonlinear applied sciences.

Occurrence and Evolution of Gaussian-Type Soliton-Like Waves in An Open Water Channel

Occurrence and Evolution of Gaussian-Type Soliton-Like Waves in An Open Water Channel

Effect of Internal Waves on Moving Small Vessels in the Sea

Internal waves are responsible for many important processes in the ocean environment (ocean ventilation, energy transfer from large-scale processes to turbulence, etc.). Our goal is to draw attention to the relatively little studied effect of internal waves on ships at sea. We encountered this effect many times while working on the study of internal waves in the shelf zone. The work was carried out from a yacht equipped with the “Rio Grande 600 kHz” ADCP, which makes it possible to measure both the parameters of internal waves and other important parameters of the medium. Two typical examples of the impact of internal waves on a yacht are given. One, when the yacht was at anchor and a train of soliton-like internal waves passed under it. The second, when the yacht was moving and passing over a train of internal waves. Internal waves passing under the moored yacht shifted its position synchronously with the periods of passing waves. A uniformly moving yacht, passing over a package of internal waves synchronously with the period of waves, alternately increased and decreased the speed of its movement. The described effect is explained by the impact on the vessel of the orbital currents of internal waves. Under our conditions, at heights of internal waves reaching 10–15 m, the ship’s speed fluctuations reached 0.30 m/s, which was more than 10% of the ship’s speed.

GENERATION OF INTERNAL WAVES BY A SUB-MESOSCALE EDDY AT CAPE SVYATOY NOS IN THE BARENTS SEA

Data on generation of soliton-like internal waves by a mesoscale eddy in the region of the Barents Sea near Cape Svyatoy Nos, widely known for its dynamic activity, are presented here. The results are based on the analysis of satellite optical (Sentinel-2 MSI) and radar (Sentinel-1 SAR) images obtained sequentially on August 8 and 9, 2021. A detailed analysis of the dynamics of the eddy and internal waves was made possible due, among other things, to the phytoplankton bloom, which created the necessary optical contrasts that make it possible to monitor the dynamics of currents in the near-surface layer of the sea. Significant parameters of recorded internal waves were measured using satellite methods and analyzed using the TPXO9.0 model and numerical solution of the internal wave equation. It has been shown, that internal waves were caused by a submesoscale eddy formed near the cape. Waves had quite low velocity (0.10–0.16 m/s) what was due to weak stratification of the water environment.

Solutions to a nonlinear Dirac–Maxwell system: from periodic waves to soliton-like waves

Using variational methods, we study the existence of stationary solutions to a periodic nonlinear Dirac–Maxwell system. Firstly, we prove the existence of n-periodic solutions to the system for all positive integers n. Then we use the concentration-compactness argument and the periodic approximation method for obtaining soliton-like solutions. We also investigate the existence of periodic ground states and soliton-like ground states, as well as their connection. Besides, results for the regularity of solutions and decay rate of the soliton-like solutions are given.

Nonlinear Soliton-like Oscillations and Waves during Geomaterial Destruction Based on Electromagnetic Radiation Signals

The work is devoted to the results of processing electromagnetic radiation signals obtained during laboratory loading of marble and diabase samples using a technique for determining the parameters of microcracks, developed and published by the author earlier. As a result of such processing, certain patterns were found in the nature of the evolution of the oscillatory process ensemble of microcracks. For example, solitary non-linear waves almost always preceded a sequence of High Frequency traces. Equations for straight lines approximating High Frequency traces in logarithmic coordinates, close to the equation of the Gutenberg–Richter law. Due to the similarity of seismic processes at different scale levels, the results of modeling at the microscale level can be used to describe seismic processes at the macroscale level, for example, to study the processes occurring immediately before destruction and at the time of destruction in order to search for repeatability and regularities. The regularities obtained can be used in the development of a predictive criterion that makes it possible to predict the time of one or another geophysical (seismic) event.

Nonlinear damped spatially periodic breathers and the emergence of soliton-like rogue waves

The spatially periodic breather solutions (SPBs) of the nonlinear Schrödinger equation, prominent in modeling rogue waves, are unstable. In this paper we numerically investigate the routes to stability of the SPBs and related rogue wave activity in the framework of the nonlinear damped higher order nonlinear Schrödinger (NLD-HONLS) equation. The NLD-HONLS solutions are initialized with single-mode and two-mode SPB data at different stages of their development. The Floquet spectral theory of the NLS equation is applied to interpret and provide a characterization of the perturbed dynamics in terms of nearby solutions of the NLS equation. A broad categorization of the routes to stability of the SPBs is determined. Novel behavior related to the effects of nonlinear damping is obtained: tiny bands of complex spectrum develop in the Floquet decomposition of the NLD-HONLS data, indicating the breakup of the SPB into either a one or two “soliton-like” structure. For solutions initialized in the early to middle stage of the development of the modulational instability, we find that all the rogue waves in the NLD-HONLS flow occur when the spectrum is in a one or two soliton-like configuration. When the solutions are initialized as the modulational instability is saturating, rogue waves also occur when the spectrum is not in a soliton-like state. Another distinctive feature of the nonlinear damped dynamics is that the growth of instabilities can be delayed and expressed at higher order due to permanent frequency downshifting.

Intercellular water exchanges trigger soliton-like waves in multicellular systems

Oscillations and waves are ubiquitous in living cellular systems. Generations of these spatiotemporal patterns are generally attributed to some mechanochemical feedbacks. Here, we treat cells as open systems, i.e., water and ions can pass through the cell membrane passively or actively, and reveal a new origin of wave generation. We show that osmotic shocks above a shock threshold will trigger self-sustained cell oscillations and result in long-range waves propagating without decrement, a phenomenon that is analogous to the excitable medium. The traveling wave propagates along the intercellular osmotic pressure gradient, and its wave speed scales with the magnitude of intercellular water flows. Furthermore, we also find that the traveling wave exhibits several hallmarks of solitary waves. Together, our findings predict a new mechanism of wave generation in living multicellular systems. The ubiquity of intercellular water exchanges implies that this mechanism may be relevant to a broad class of systems.

Harnessing Cascading Dominoes for Peristaltic Wave Motion

Organisms have evolved many simple yet effective muscular movements. Of these, peristaltic wave motion is among the most prolific and can be found across organism scales and in a variety of organs that transport objects or provide locomotion. Several robotic platforms have mimicked peristaltic movement, which resembles soliton propagation; however, the complexity of integrating multiple sequential actuators and the fragility of these systems have limited their practical application. In this work, we demonstrate the potential of cascading dominoes to realize soliton-like wave movements that can transport objects and locomote with only two actuators. Unlike conventional domino series where falling dominoes produce a soliton at the collapsing frontier, we added a revolute joint on each domino to ensure all the dominoes stack continuously, which inherently changes the soliton wave shape. We show how domino geometry determines the wave shape of the soliton, both analytically and experimentally, and how actuator speed affects the wave speed. Inspired by peristaltic wave motion in living organisms, we used the cascading dominoes to move objects through an artificial esophageal conduit and build a “mantis shrimp” crawler. Cascading dominoes enable a new type of actuator mechanism and provide new opportunities for robotic motion.

Cross-scale dynamic interactions in compacting porous media as a trigger to pattern formation

SUMMARYPatterns in nature are often interpreted as a product of reaction-diffusion processes which result in dissipative structures. Thermodynamic constraints allow prediction of the final state but the dynamic evolution of the microprocesses is hidden. We introduce a new microphysics-based approach that couples the microscale cross-constituent interactions to the large-scale dynamic behaviour, which leads to the discovery of a family of soliton-like excitation waves. These waves can appear in hydromechanically coupled porous media as a reaction to external stimuli. They arise, for instance, when mechanical forcing of the porous skeleton releases internal energy through a phase change, leading to tight coupling of the pressure in the solid matrix with the dissipation of the pore fluid pressure. In order to describe these complex multiscale interactions in a thermodynamic consistent framework, we consider a dual-continuum system, where the large-scale continuum properties of the matrix–fluid interaction are described by a reaction-self diffusion formulation, and the small-scale dissipation of internal energy by a reaction-cross diffusion formulation that spells out the macroscale reaction and relaxes the adiabatic constraint on the local reaction term in the conventional reaction-diffusion formalism. Using this approach, we recover the familiar Turing bifurcations (e.g. rhythmic metamorphic banding), Hopf bifurcations (e.g. Episodic Tremor and Slip) and present the new excitation wave phenomenon. The parametric space is investigated numerically and compared to serpentinite deformation in subduction zones.

Soliton-like surface waves propagating along the interface between a metamaterial and an antiferromagnet

Nonlinear surface waves propagating along a single interface between a linear metamaterial and a nonlinear antiferromagnet are theoretically studied. The frequency response of transverse-magnetic-polarized waves covering the frequency range from the microwave to the infrared regime is investigated in detail. We find that the waveguide supports soliton-like surface waves with low-pass response in the microwave range and bandpass response in the microwave and infrared ranges. The bandwidth is a function of the nonlinearity and can be tuned by controlling the incident wave power.

Abundant explicit non-traveling wave solutions for the (2+1)-dimensional breaking soliton equation

By combining the generalized variable separation method with the extended homoclinic test approach (EHTA) explicit exact non-traveling wave solutions of the (2+1)-dimensional breaking soliton equation are constructed. With the aid of symbolic computation, a series of new non-traveling wave solutions of the (2+1)-dimensional breaking soliton equation are expressed explicitly. These non-traveling wave solutions are new solutions with three arbitrary functions, which have a more general form than that in the previous literatures. The result obtained here reveals the complex structure of the solutions of the (2+1)-dimensional breaking soliton equation. The previous results obtained in literatures can be regarded as special cases here. When some arbitrary functions included in these solutions are taken as some special functions, exact periodic solitary wave, cross soliton-like wave, periodic cross-kink wave, periodic two-solitary wave are presented.

Dynamics of a localized ring nonlinear wave in a carbon nanotube

The propagation of nonlinear localized soliton-like super-acoustic longitudinal waves in a carbon nanotube was explored by the molecular dynamics modeling. Ring waves were excited by pulsed action on all the atoms of the edge zigzag layer and they got an initial velocity along the axis of the nanotube. It was shown that the velocity of a localized (soliton-like) ring super-acoustic wave increased with an increase in the initial velocity of the edge atoms. The calculated dependence of the propagating wave velocity on the initial velocity of the edge atoms was given. Keywords: carbon nanotube, ring nonlinear waves, molecular dynamics.