The integrable Vakhnenko–Parkes (VP) and the modified Vakhnenko–Parkes (MVP) equations: Multiple real and complex soliton solutions

In this work, we show that the integrable Vakhnenko–Parkes (VP) equation passes the Painlevé test and admits multiple real and multiple complex soliton solutions. We also present, for the first time, the modified Vakhnenko-Parkes (MVP) equation, show its complete integrability, and formally derive its multiple real and multiple complex soliton solutions. To achieve the goal set for this work, we introduce two complex forms of the simplified Hirota’s method, the first works effectively for the VP equation, and the other form is nicely applicable for the MVP equation. We believe that establishing the complex forms will shed light on complex solitons of other integrable equations.     

Exact solutions of discrete complex cubic-quintic Ginzburg-Landau equation with non-local quintic term

In this paper, via the extended tanh-function approach, the abundant exact solutions for discrete complex cubic-quintic Ginzburg-Landau equation, including chirpless bright soliton, chirpless dark soliton, constant magnitude solution (plane wave solution), triangular function solutions and some solutions with alternating phases, etc. are obtained. Meanwhile, the range of parameters where some exact solution exist are given. Among these solutions, solutions with alternating phases do not have continuous analogs. Moreover, in the lattice, the points of singularity of tan-type and sec-type solutions can be ‘between sites’ and thus the singularities can be avoided.     

The （w/g）-expansion method and its application to Vakhnenko equation

This paper presents a new function expansion method for finding travelling wave solutions of a nonlinear evolution equation and calls it the (ω/g)-expansion method, which can be thought of as the generalization of (G /G)-expansion given by Wang et al recently. As an application of this new method, we study the well-known Vakhnenko equation which describes the propagation of high-frequency waves in a relaxing medium. With two new expansions, general types of soliton solutions and periodic solutions for Vakhnenko equation are obtained.     

The Rayleigh-Plesset equation in terms of volume with explicit shear losses

The most common nonlinear equation of motion for the damped pulsation of a spherical gas bubble in an infinite body of liquid is the Rayleigh-Plesset equation, expressed in terms of the dependency of the bubble radius on the conditions pertaining in the gas and liquid (the so-called ‘radius frame’). However over the past few decades several important analyses have been based on a heuristically derived small-amplitude expansion of the Rayleigh-Plesset equation which considers the bubble volume, instead of the radius, as the parameter of interest, and for which the dissipation term is not derived from first principles. So common is the use of this equation in some fields that the inherent differences between it and the ‘radius frame’ Rayleigh-Plesset equation are not emphasised, and it is important in comparing the results of the two equations to understand that they differ both in terms of damping, and in the extent to which they neglect higher order terms. This paper highlights these differences. Furthermore, it derives a ‘volume frame’ version of the Rayleigh-Plesset equation which contains exactly the same basic physics for dissipation, and retains terms to the same high order, as does the ‘radius frame’ Rayleigh-Plesset equation. Use of this equation will allow like-with-like comparisons between predictions in the two frames.     

Nonlinear propagation of vector extremely short pulses in a medium of symmetric and asymmetric molecules

The nonlinear propagation of extremely short electromagnetic pulses in a medium of symmetric and asymmetric molecules placed in static magnetic and electric fields is theoretically studied. Asymmetric molecules differ in that they have nonzero permanent dipole moments in stationary quantum states. A system of wave equations is derived for the ordinary and extraordinary components of pulses. It is shown that this system can be reduced in some cases to a system of coupled Ostrovsky equations and to the equation intagrable by the method for an inverse scattering transformation, including the vector version of the Ostrovsky–Vakhnenko equation. Different types of solutions of this system are considered. Only solutions representing the superposition of periodic solutions are single-valued, whereas soliton and breather solutions are multivalued.     

Note on wavefront dislocation in surface water waves

At singular points of a wave field, where the amplitude vanishes, the phase may become singular and wavefront dislocation may occur. In this Letter we investigate for wave fields in one spatial dimension the appearance of these essentially linear phenomena. We introduce the Chu-Mei quotient as it is known to appear in the ‘nonlinear dispersion relation’ for wave groups as a consequence of the nonlinear transformation of the complex amplitude to real phase-amplitude variables. We show that unboundedness of this quotient at a singular point, related to unboundedness of the local wavenumber and frequency, is a generic property and that it is necessary for the occurrence of phase singularity and wavefront dislocation, while these phenomena are generic too. We also show that the ‘soliton on finite background’, an explicit solution of the NLS equation and a model for modulational instability leading to extreme waves, possesses wavefront dislocations and unboundedness of the Chu-Mei quotient.     

Solitonic axion condensates modeling dark matter halos

Instead of fluid type dark matter (DM), axion-like scalar fields with a periodic self-interaction or some truncations of it are analyzed as a model of galaxy halos. It is probed if such cold Bose–Einstein type condensates could provide a viable soliton type interpretation of the DM ‘bullets’ observed by means of gravitational lensing in merging galaxy clusters. We study solitary waves for two self-interacting potentials in the relativistic Klein–Gordon equation, mainly in lower dimensions, and visualize the approximately shape-invariant collisions of two ‘lump’ type solitons.     

Breather‐to‐soliton transitions and nonlinear wave interactions for the nonlinear Schrödinger equation with the sextic operators in optical fibers

We find that the sextic nonlinear Schrödinger (NLS) equation admits breather‐to‐soliton transitions. With the Darboux transformation, analytic breather solutions with imaginary eigenvalues up to the second order are explicitly presented. The condition for breather‐to‐soliton transitions is explicitly presented and several examples of transitions are shown. Interestingly, we show that the sextic NLS equation admits not only the breather‐to‐bright‐soliton transitions but also the breather‐to‐dark‐soliton transitions. We also show the interactions between two solitons on the constant backgrounds, as well as between breather and soliton.     

Analysis of car-following model considering driver’s physical delay in sensing headway

An extended car-following model is proposed by taking into account the delay of the driver’s response in sensing headway. The stability condition of this model is obtained by using the linear stability theory. The results show that the stability region decreases when the driver’s physical delay in sensing headway increases. The KdV equation and mKdV equation near the neutral stability line and the critical point are respectively derived by applying the reductive perturbation method. The traffic jams could be thus described by soliton solution and kink-antikink soliton solution for the KdV equation and mKdV equation respectively. The numerical results in the form of the space-time evolution of headway show that the stabilization effect is weakened when the driver’s physical delay increases. It confirms the fact that the delay of driver’s response in sensing headway plays an important role in jamming transition, and the numerical results are in good agreement with the theoretical analysis.     

Multidimensional magnesium oxide nanostructures with cone-shaped branching

Branching structures in nanometer level are of great importance in developing nanoscale science and functional electrical devices. In this letter, multidimensional magnesium oxide structures with cone-shaped branching have been mass-produced using a simple chemical vapor deposition method. The dominant structures in the product include two-dimensional ‘+’, ‘T’, or ‘Γ’ assemblies, and three-dimensional complex configurations. The results presented here enrich the nanoscale community with new basic materials for the fabrication of functional electrical and chemical sensing devices.     

Drift ion acoustic solitons in an inhomogeneous 2-D quantum magnetoplasma

Linear and nonlinear propagation characteristics of quantum drift ion acoustic waves are investigated in an inhomogeneous two-dimensional plasma employing the quantum hydrodynamic (QHD) model. In this regard, the dispersion relation of the drift ion acoustic waves is derived and limiting cases are discussed. In order to study the drift ion acoustic solitons, nonlinear quantum Kadomstev-Petviashvilli (KP) equation in an inhomogeneous quantum plasma is derived using the drift approximation. The solution of quantum KP equation using the tangent hyperbolic (tanh) method is also presented. The variation of the soliton with the quantum Bohm potential, the ratio of drift to soliton velocity in the co-moving frame, , and the increasing magnetic field are also investigated. It is found that the increasing number density decreases the amplitude of the soliton. It is also shown that the fast drift soliton (i.e., v_*>u) decreases whereas the slow drift soliton (i.e., v_*<u) increases the amplitude of the soliton. Finally, it is shown that the increasing magnetic field increases the amplitude of the quantum drift ion acoustic soliton. The stability of the quantum KP equation is also investigated. The relevance of the present investigation in dense astrophysical environments is also pointed out.     

Canonical quantization of Galilean covariant field theories

The Galilean-invariant field theories are quantized by using the canonical method and the five-dimensional Lorentz-like covariant expressions of non-relativistic field equations. This method is motivated by the fact that the extended Galilei group in 3 + 1 dimensions is a subgroup of the inhomogeneous Lorentz group in 4 + 1 dimensions. First, we consider complex scalar fields, where the Schrödinger field follows from a reduction of the Klein-Gordon equation in the extended space. The underlying discrete symmetries are discussed, and we calculate the scattering cross-sections for the Coulomb interaction and for the self-interacting term λΦ⁴. Then, we turn to the Dirac equation, which, upon dimensional reduction, leads to the Lévy-Leblond equations. Like its relativistic analogue, the model allows for the existence of antiparticles. Scattering amplitudes and cross-sections are calculated for the Coulomb interaction, the electron-electron and the electron-positron scattering. These examples show that the so-called ‘non-relativistic’ approximations, obtained in low-velocity limits, must be treated with great care to be Galilei-invariant. The non-relativistic Proca field is discussed briefly.     

Determination of the timing properties of a pulsed positron lifetime beam by the application of an electron gun and a fast microchannel plate

We show that the timing properties of a pulsed low-energy positron lifetime beam can be conveniently tested by an electron beam. We apply this method to study the time resolution of the beam and electron scattering in flat and ‘sawtooth’ shaped choppers. The results show that (i) time resolution of 160 ps is obtained, (ii) the scattering of the electrons and the secondary electron yield of the flat chopper make the time resolution worse and background poor, and (iii) both these problems can be solved by using a ‘sawtooth’ shaped chopper. We also compare these results to beam simulations.     

On one-port characterization of noise sources in ducts by using external loads

Equivalent acoustic source characterization of duct-borne fluid machinery noise is often undertaken by interpolating the results of two-microphone pressure measurements with different external acoustic loads over a linear one-port source model. If the source is time-invariant, the one-port source characteristics can be determined by using only two external loads. This is well known as the two-load method. An extension of the two-load method for time-variant sources is also available and known as the multiple-load method. In these methods the source is treated as a ‘black-box’. This paper addresses the problem of one-port source characterization when the linear operations inherent in the ‘black-box’ are known explicitly. The equations governing the explicit one-port source models are derived and the source characteristics are shown to be measurable using only few acoustic loads. It is not the purpose of this paper to discuss the application of these models to any specific fluid machinery; however, of particular interest are the explicit source models that require only two loads. Numerical results are presented to show some features of such time-invariant and time-variant explicit one-port source models.     

Vandermonde-like determinants’ representations of Darboux transformations and explicit solutions for the modified Kadomtsev-Petviashvili equation

Recently, the (2+1)-dimensional modified Kadomtsev-Petviashvili (mKP) equation was decomposed into two known (1+1)-dimensional soliton equations by Dai and Geng [H.H. Dai, X.G. Geng, J. Math. Phys. 41 (2000) 7501]. In the present paper, a systematic and simple method is proposed for constructing three kinds of explicit N-fold Darboux transformations and their Vandermonde-like determinants’ representations of the two known (1+1)-dimensional soliton equations based on their Lax pairs. As an application of the Darboux transformations, three explicit multi-soliton solutions of the two (1+1)-dimensional soliton equations are obtained; in particular six new explicit soliton solutions of the (2+1)-dimensional mKP equation are presented by using the decomposition. The explicit formulas of all the soliton solutions are also expressed by Vandermonde-like determinants which are remarkably compact and transparent.     

Considerations to Rayleigh’s hypothesis

In 1907 Lord Rayleigh published a paper on the dynamic theory of gratings. In this paper he presented a rigorous approach for solving plane wave scattering on periodic surfaces. Moreover he derived explicit expressions for a perfectly conducting sinusoidal surface, and for perpendicular incidence of the electromagnetic plane wave. This paper was criticized by Lippmann in 1953 for he assumed Rayleigh’s approach to be incomplete. Since this time there have been published several arguments, proofs, and discussions concerning the correctness and the range of validity of Rayleigh’s approach not only for plane wave scattering on gratings but also for light scattering on nonspherical structures, in general. In the paper at hand we will discuss the different point of views on what is called “Rayleigh’s hypothesis” as well as the relevance of a found theoretical limit for its validity. Furthermore we present a numerical treatment of the original scattering problem of a p-polarized plane wave perpendicularly incident on a perfectly conducting sinusoidal surface (i.e., the scalar Dirichlet problem). In doing so we emphasizes the near-field solution especially within the grooves of the grating up to points on the surface, and below the surface. Two different Green’s function formulations of Huygens’ principle are used as starting points. One of this formulation results in the general T-matrix approach which is considered to be affected by Rayleigh’s hypothesis especially for near-field calculations. The other formulation provides a conventional boundary integral equation which is in accordance with Lippmann’s point of view and free of problems with Rayleigh’s hypothesis. But the obtained results show that Lippmann’s argumentation do not withstand a critical numerical analysis, and that the independence of least-squares approaches from Rayleigh’s hypothesis, as understood and proven by Millar, seems to hold also for certain methods which does not fit into such an approach.     

Cross-correlation dynamics in financial time series

The dynamics of the equal-time cross-correlation matrix of multivariate financial time series is explored by examination of the eigenvalue spectrum over sliding time windows. Empirical results for the S&P 500 and the Dow Jones Euro Stoxx 50 indices reveal that the dynamics of the small eigenvalues of the cross-correlation matrix, over these time windows, oppose those of the largest eigenvalue. This behaviour is shown to be independent of the size of the time window and the number of stocks examined.A basic one-factor model is then proposed, which captures the main dynamical features of the eigenvalue spectrum of the empirical data. Through the addition of perturbations to the one-factor model, (leading to a ‘market plus sectors’ model), additional sectoral features are added, resulting in an Inverse Participation Ratio comparable to that found for empirical data. By partitioning the eigenvalue time series, we then show that negative index returns, (drawdowns), are associated with periods where the largest eigenvalue is greatest, while positive index returns, (drawups), are associated with periods where the largest eigenvalue is smallest. The study of correlation dynamics provides some insight on the collective behaviour of traders with varying strategies.     

Thermodynamics of relation-based systems with applications in econophysics,sociophysics, and music

A methodology was developed to analyze relation-based systems evolving in time by using the fundamental concepts of thermodynamics. The behavior of such systems can be tracked from the scattering matrix which is actually a network of directed vectors (or pathways) connecting subsequent values, which characterize an event, such as the index values in stock markets. A system behaves in a rigid (elastic) way to an external effect and resists permanent deformation, or it behaves in a viscous (or soft) way and deforms in an irreversible way. It was shown in the past that a formula derived using the slope of paths gives a measure about the extent of viscoelastic behavior of relation-based systems Gündüz (2009) [5] Gündüz and Gündüz (2010) [6]. In this research the ‘work’ associated with ‘elastic’ component, and ‘heat’ associated with ‘viscous’ component were discussed and elaborated. In a simple two subsequent pathway system in a scattering diagram the first vector represents ‘the cause’ and the second ‘the effect’. By using work and heat energy relations that involve force and also storage and loss modulus terms, respectively, one can calculate the energy involved in relation-based systems. The modulus values can be found from the parallel and vertical components of the second vector with respect to the first vector. Once work-like and heat-like terms were determined the internal energy is also easily found from their summation. The parallel and vertical components can also be used to calculate the magnitude of torque and torque energy in the system. Three cases, (i) the behavior of the NASDAQ-100 index, (ii) a social revolt, and (iii) the structure of a melody were analyzed for their ‘work-like’, ‘heat-like’, and ‘torque-like’ energies in the course of their evolution. NASDAQ-100 exhibits highly dissipative behavior, and its work terms are very small but heat terms are of large magnitude. Its internal energy highly fluctuates in time. In the social revolt studied work and heat terms are of comparable magnitude. The melody depicts highly organized structure, and usually has larger work terms than heat terms, but at some intervals heat terms burst out and attain very large magnitudes. Torque terms reach high values when the system is recovering from a minimum value.     

On the destabilizing effect of damping on discrete and continuous circulatory systems

The ‘Ziegler paradox’, concerning the destabilizing effect of damping on elastic systems loaded by nonconservative positional forces, is addressed. The paper aims to look at the phenomenon in a new perspective, according to which no surprising discontinuities in the critical load exist between undamped and damped systems. To show that the actual critical load is found as an (infinitesimal) perturbation of one of the infinitely many sub-critically loaded undamped systems. A series expansion of the damped eigenvalues around the distinct purely imaginary undamped eigenvalues is performed, with the load kept as a fixed, although unknown, parameter. The first sensitivity of the eigenvalues, which is found to be real, is zeroed, so that an implicit expression for the critical load multiplier is found, which only depends on the ‘shape’ of damping, being independent of its magnitude. An interpretation is given of the destabilization paradox, by referring to the concept of ‘modal damping’, according to which the sign of the projection of the damping force on the eigenvector of the dual basis, and not on the eigenvector itself, is the true responsible for stability. The whole procedure is explained in detail for discrete systems, and successively extended to continuous systems. Two sample structures are studied for illustrative purposes: the classical reverse double-pendulum under a follower force and a linear visco-elastic beam under a follower force and a dead load.     

Exotic electron states and tunable magneto-transport in a fractal Aharonov–Bohm interferometer

A Sierpinski gasket fractal network model is studied in respect of its electronic spectrum and magneto-transport when each ‘arm’ of the gasket is replaced by a diamond shaped Aharonov–Bohm interferometer, threaded by a uniform magnetic flux. Within the framework of a tight binding model for non-interacting, spinless electrons and a real space renormalization group method we unravel a class of extended and localized electronic states. In particular, we demonstrate the existence of extreme localization of electronic states at a special finite set of energy eigenvalues, and an infinite set of energy eigenvalues where the localization gets ‘delayed’ in space (staggered localization). These eigenstates exhibit a multitude of localization areas. The two terminal transmission coefficient and its dependence on the magnetic flux threading each basic Aharonov–Bohm interferometer is studied in details. Sharp switch on–switch off effects that can be tuned by controlling the flux from outside, are discussed. Our results are analytically exact.