Stationary Solitary Waves Research Articles

Qualitative difference between large waves in deep and shallow fluid formulations

The note addresses a qualitative difference between shallow (vertically confined) and deep (vertically non-confined) fluid geometries for stationary internal solitary waves. It is shown that in a deep fluid, the propagation velocity of large amplitude wave (with a vortex inside) is greater than the velocity predicted by small but finite amplitude theory, known as the Benjamin-Ono model. This effect has been found both asymptotically and experimentally. For the case of a shallow fluid, the situation is qualitatively different. The speed of a wave with vortex inside is smaller than that predicted by the Korteweg-de Vries theory. The reported observation could distinguish wave motions in shallow (confined) and deep (non-confined) geometries and seems to be important in a variety of applications.

Properties of the generalized Chavy-Waddy–Kolokolnikov model for description of bacterial colonies

The Chavy-Waddy–Kolokolnikov model with dispersion for describing bacterial colonies is considered. This mathematical model is described by a nonlinear partial differential equation of the fourth order. This equation does not pass the Painlevé test and the Cauchy problem cannot be solved by the inverse scattering transform. Some new properties of the Chavy-Waddy–Kolokolnikov model are studied. Analytical solutions of the equation in traveling wave variables are found taking into account the dispersion coefficient. It is shown that, unlike the model without dispersion, a bacterial cluster can move, which allows us to consider dispersion as some kind of control for bacterial colony. Using numerical modeling, we also demonstrate that the initial concentration of bacteria in the form of a random distribution over time transforms into a periodic wave, followed by a transition to a stationary solitary wave without taking dispersion into account.

Propagation of nonlinear waves in multi-component pair plasmas and electron–positron–ion plasmas

The propagation of small amplitude stationary profile nonlinear solitary waves in a pair plasma is investigated by employing the reductive perturbation technique via the well-known Korteweg–de Vries (KdV) and modified KdV (mKdV) equations. This study tends to derive the exact form of nonlinear solutions and study their characteristics. Two distinct pair-ion species of opposite polarity and the same mass are considered in addition to a massive charged background species that is assumed to be stationary, and given the frequency scale of interest within the pair-ion context, the third species is thought of as a background defect (e.g., charged dust) component. On the opposite hand, the model conjointly applies formally to electron–positron–ion plasmas if one neglects electron–positron annihilation. A parametric analysis is carried out, with regard to the impact of the dusty plasma composition (background number density), species temperature(s), and background species. It is seen that distinguishable solitary profiles are observed for KdV and mKdV equations. The results are connected in pair-ion (fullerene) experiments and potentially in astrophysical environments of Halley’s comet and pulsar magnetosphere as well.

Discrete embedded solitary waves and breathers in one-dimensional nonlinear lattices

For a one-dimensional linear lattice, earlier work has shown how to systematically construct a slowly- decaying linear potential bearing a localized eigenmode embedded in the continuous spectrum. Here, we extend this idea in two directions: The first one is in the realm of the discrete nonlinear Schrödinger, where the linear operator of the Schrödinger type is considered in the presence of a Kerr focusing or defocusing nonlinearity and the embedded linear mode is continued into the nonlinear regime as a discrete solitary wave. The second case is the Klein-Gordon setting, where the presence of a cubic nonlinearity leads to the emergence of embedded-in-the-continuum discrete breathers. In both settings, it is seen that the stability of the modes near the linear limit turns into instability as nonlinearity is increased past a critical value, leading to a dynamical delocalization of the solitary wave (or breathing) state. Finally, we suggest a concrete experiment to observe these embedded modes using a bi-inductive electrical lattice.

Oblique stationary solitary waves in turbulent free-surface flow

It is shown that an oblique solitary wave may be caused by an oblique strip of enlarged bottom roughness. The analysis is valid for very large Reynolds numbers and very small slopes. Considered are strips with a constant width in the longitudinal direction and with constant roughness. For a strip whose length is much larger than its width, the equations of motion can be simplified by assuming a three-dimensional flow that is independent of the coordinate in the longitudinal direction of the strip. Furthermore, it is assumed that the Froude number in terms of the velocity component normal to the strip is slightly larger than the critical value 1. This facilitates application of an asymptotic expansion method developed previously for plane flow. The analysis does not require modeling of the Reynolds stresses. It turns out that the velocity profile of the three-dimensional flow is not skewed in the first order. The shape, amplitude, and position of the oblique stationary wave are obtained from solutions of the steady-state version of an extended Korteweg-de-Vries equation. In general, the oblique wave is a classical solitary wave with a long, but shallow tail. However, tails are missing for certain combinations of Froude number and parameters characterizing the strip. Measurements and numerical solutions of the full Reynolds equations are already available for orthogonal solitary waves, lending support to the results of the asymptotic analysis. In addition, a novel boundary condition at the free surface is given, and the velocity distribution in fully developed flow is determined.

Steady Solitary and Periodic Waves in a Nonlinear Nonintegrable Lattice

In this paper, stationary solitary and periodic waves of a nonlinear nonintegrable lattice are numerically constructed using a two-stage approach. First, as a result of continualization, a nonintegrable generalized Boussinesq—Ostrovsky equation is obtained, for which the solitary-wave and periodic solutions are numerically found by the Petviashvili method. In the second stage, discrete analogs of the obtained solutions are used as initial conditions in the numerical simulation of the original lattice. It is shown that the initial perturbations constructed in this way propagate along the lattice without changing their shape.

Nondestructive Evaluation of Solids Based on Deformation Wave Theory

The application of a recent field theory of deformation and fracture to nondestructive testing (NDT) is discussed. Based on the principle known as the symmetry of physical laws, the present field theory formulates all stages of deformation including the fracturing stage on the same theoretical basis. The formalism derives wave equations that govern the spatiotemporal characteristics of the differential displacement field of solids under deformation. The evolution from the elastic to the plastic stage of deformation is characterized by a transition from longitudinal (compression) wave to decaying longitudinal/transverse wave characteristics. The evolution from the plastic to the fracturing stage is characterized by transition from continuous wave to solitary wave characteristics. Further, the evolution from the pre-fracturing to the final fracturing stage is characterized by transition from the traveling solitary wave to stationary solitary wave characteristics. In accordance with these transitions, the criterion for deformation stage is defined as specific spatiotemporal characteristics of the differential displacement field. The optical interferometric technique, known as Electronic Speckle-Pattern Interferometry (ESPI), is discussed as an experimental tool to visualize those wave characteristics and the associated deformation-stage criteria. The wave equations are numerically solved for the elastoplastic stages, and the resultant spatiotemporal behavior of the differential displacement field is compared with the experimental results obtained by ESPI. Agreement between the experimental and numerical results validates the present methodology at least for the elastoplastic stages. The solitary wave characteristics in the fracturing stages is discussed based on the experimental results and dislocation theory.

Nonlinear Waves in a Rotating Ocean (The Ostrovsky Equation and Its Generalizations and Applications)

This review presents theoretical, numerical, and experimental results of a study of the structure and dynamics of weakly nonlinear internal waves in a rotating ocean accumulated over the past 40 years since the time when the approximate equation, called the Ostrovsky equation, was derived in 1978. The relationship of this equation with other well-known wave equations, the integrability of the Ostrovsky equation, and the condition for the existence of stationary solitary waves and envelope solitary waves are discussed. The adiabatic dynamics of Korteweg–de Vries solitons in the presence of fluid rotation is described. The mutual influence of the ocean inhomogeneity and rotation effect on the dynamics of solitary waves is considered. The universality of the Ostrovsky equation as applied to waves in other media (solids, plasma, quark-gluon plasma, and optics) is noted.

Electroacoustic Waves in a Collision-Free Magnetized Superthermal Bi-Ion Plasma

The electroacoustic waves, particularly ion-acoustic waves (IAWs), and their expansion in the medium of a magnetized collision-free plasma system has been investigated theoretically. The plasma system is assumed to be composed of both positively and negatively charged mobile ion species and kappa-distributed hot electron species. In the nonlinear perturbation regime, the magnetized Korteweg–de Vries (KdV) and magnetized modified KdV (mKdV) equations are derived by using reductive perturbation method. The prime features (i.e., amplitude, phase speed, width, etc.) of the IAWs are studied precisely by analyzing the stationary solitary wave solutions of the magnetized KdV and magnetized mKdV equations, respectively. It occurs that the basic properties of the IAWs are significantly modified in the presence of the excess superthermal hot electrons, obliqueness, the plasma particle number densities, etc. It is also observed that, in case of magnetized KdV solitary waves, both compressive and rarefactive structures are formed, whereas only compressive structures are found for the magnetized mKdV solitary waves. The implication of our results in some space and laboratory plasma situations is concisely discussed.

Near‐critical turbulent open‐channel flow over wavy bottom

AbstractSteady two‐dimensional turbulent free‐surface flow in a channel with mild baseline slope is considered. The shape of the channel bottom is assumed to be undular with a very small amplitude. Asymptotic expansions for large Reynolds numbers and Froude numbers slightly above the critical value 1, respectively, give for the surface elevation a differential equation of KdV‐type, with the additional terms representing turbulent dissipation and forcing due to the wavy bottom, respectively. No turbulence modelling is required. This asymptotic approach was used in [1] to describe stationary solitary waves in a channel with plain bottom and small variations in the bottom friction coefficient. It was shown recently [2,3] that there exist stationary single‐wave solutions of a different kind that are characterized by smaller wave amplitudes. In [4], both kinds of single stationary waves above single obstacles (bumps, ramps) are investigated theoretically and experimentally. In this paper, stationary space‐periodic surface waves for channel bottoms with undular shape are studied. First, a one‐parametric family of exact solutions for particular bottom shapes is derived. Remarkably, these particular solutions exist only in a narrow parameter range. The solutions are reproduced with a numerical solver to verify that the solver gives correct results. Secondly, a different type of asymptotic expansion is performed in order to describe solutions characterized by smaller wave amplitudes. The resulting linear differential equation is solved numerically. Both solutions are briefly discussed.

Continuous families of solitary waves in non-symmetric complex potentials: A Melnikov theory approach

The existence of stationary solitary waves in symmetric and non-symmetric complex potentials is studied by means of Melnikov’s perturbation method. The latter provides analytical conditions for the existence of such waves that bifurcate from the homogeneous nonlinear modes of the system and are located at specific positions with respect to the underlying potential. It is shown that the necessary conditions for the existence of continuous families of stationary solitary waves, as they arise from Melnikov theory, provide general constraints for the real and imaginary part of the potential, that are not restricted to symmetry conditions or specific types of potentials. Direct simulations are used to compare numerical results with the analytical predictions, as well as to investigate the propagation dynamics of the solitary waves.

Electrostatic Solitary Pulses in a Dusty Electronegative Magnetoplasma

The nonlinear characteristics of dust-electron-acoustic (DEA) waves in a dusty electronegative magnetoplasma system consisting of nonextensive hot electrons, inertial cold electrons, positively charged static ions, and negatively charged immobile dust grains has been investigated. In this observation, the well-known reductive perturbation technique is employed to determine different types of nonlinear dynamical equations, namely, magnetized Korteweg–de Vries (KdV), magnetized modified KdV (mKdV), and magnetized Gardner equations. The stationary solitary wave and double layer solution of these three equations, which describe the characteristics of solitary waves and double layers of DEA waves, are obtained and numerically analyzed. It is noticed that various plasma parameters (viz., hot electron nonextensivity, positive ion-to-cold electron number density ratio, dust-to-cold electron number density ratio, etc.) significantly affect the basic properties of DEA solitary waves (DEASWs) and Gardner solitons (GSs). The prodigious results found from this theoretical investigation may be useful for researchers to investigate the nonlinear structures in various space and laboratory plasmas.

Nonlinear standing waves in an array of coherently coupled Bose-Einstein condensates

Stationary solitary waves are studied in an array of $M$ linearly-coupled one-dimensional Bose-Einstein condensates (BECs) by means of the Gross-Pitaevskii equation. Solitary wave solutions with the character of overlapping dark solitons, Josephson vortex - antivortex arrays, and arrays of half-dark solitons are constructed for $M>2$ from known solutions for two coupled BECs. Additional solutions resembling vortex dipoles and rarefaction pulses are found numerically. Stability analysis of the solitary waves reveals that overlapping dark solitons can become unstable and susceptible to decay into arrays of Josephson vortices. The Josephson vortex arrays have mixed stability but for all parameters we find at least one stationary solitary wave configuration that is dynamically stable. The different families of nonlinear standing waves bifurcate from one another. In particular we demonstrate that Josephson-vortex arrays bifurcate from dark soliton solutions at instability thresholds. The stability thresholds for dark soliton and Josephson-vortex type solutions are provided, suggesting the feasibility of realization with optical lattice experiments.

Near-critical turbulent open-channel flows over bumps and ramps

Steady two-dimensional turbulent free-surface flow in a channel with a slightly uneven bottom is considered. The shape of the unevenness of the bottom can be in the form of a bump or a ramp of very small height. The slope of the channel bottom is assumed to be small, and the bottom roughness is assumed to be constant. Asymptotic expansions for very large Reynolds numbers and Froude numbers close to the critical value {Fr} = 1, respectively, are performed. The relative order of magnitude of two small parameters, i.e. the bottom slope and ({Fr}-1), is defined such that no turbulence modelling is required. The result is a steady-state version of an extended Korteweg–de Vries equation for the surface elevation. Other flow quantities, such as pressure, flow velocity components, and bottom shear stress, are expressed in terms of the surface elevation. An exact solution describing stationary solitary waves of the classical shape is obtained for a bottom of a particular shape. For more general shapes of ramps and bumps, stationary solitary waves of the classical shape are also obtained as a first approximation in the limit of small, but nonzero, dissipation. With the exception of an eigensolution for a ramp, an outer region has to be introduced. The outer solution describes a ’tail’ that is attached to the stationary solitary wave. In addition to the solutions of the solitary-wave type, solutions of smaller amplitudes are obtained both numerically and analytically. Experiments in a water channel confirm the existence of both types of stationary single waves.

One- and Two-dimensional Solitary Wave States in the Nonlinear Kramers Equation with Movement Direction as a Variable

We study self-propelled particles by direct numerical simulation of the nonlinear Kramers equation for self-propelled particles. In our previous paper, we studied self-propelled particles with velocity variables in one dimension. In this paper, we consider another model in which each particle exhibits directional motion. The movement direction is expressed with a variable $\phi$. We show that one-dimensional solitary wave states appear in direct numerical simulations of the nonlinear Kramers equation in one- and two-dimensional systems, which is a generalization of our previous result. Furthermore, we find two-dimensionally localized states in the case that each self-propelled particle exhibits rotational motion. The center of mass of the two-dimensionally localized state exhibits circular motion, which implies collective rotating motion. Finally, we consider a simple one-dimensional model equation to qualitatively understand the formation of the solitary wave state.

Spectral stability of the bifurcation state of an arterial model with perivascular soft tissues

We model an artery with perivascular soft tissue as a uniform cylindrical membrane tube surrounded by a flexible substrate with distributed stiffness. We derive the equations of motion of the arterial model, and obtain evolution equation derived in the long wavelength limit from the general equations of motion. We analyze the stability of axisymmetric perturbations at the bifurcation state taking into the consideration of surrounding soft tissue stiffness and constant axial stretch. We observe that the surrounding soft tissues progressively reduce the domain of real valued solutions with increasing constant axial stretch. The results suggest that the stationary solitary wave solution is unstable.

Solitary wave for a nonintegrable discrete nonlinear Schrödinger equation in nonlinear optical waveguide arrays**Project supported by the National Natural Science Foundation of China (Grant Nos. 11671255 and 11701510), the Ministry of Economy and Competitiveness of Spain (Grant No. MTM2016-80276-P (AEI/FEDER, EU)), and the China Postdoctoral Science Foundation (Grant No. 2017M621964).

We study a nonintegrable discrete nonlinear Schrödinger (dNLS) equation with the term of nonlinear nearest-neighbor interaction occurred in nonlinear optical waveguide arrays. By using discrete Fourier transformation, we obtain numerical approximations of stationary and travelling solitary wave solutions of the nonintegrable dNLS equation. The analysis of stability of stationary solitary waves is performed. It is shown that the nonlinear nearest-neighbor interaction term has great influence on the form of solitary wave. The shape of solitary wave is important in the electric field propagating. If we neglect the nonlinear nearest-neighbor interaction term, much important information in the electric field propagating may be missed. Our numerical simulation also demonstrates the difference of chaos phenomenon between the nonintegrable dNLS equation with nonlinear nearest-neighbor interaction and another nonintegrable dNLS equation without the term.

Stationary single waves in turbulent open‐channel flow

AbstractSteady two‐dimensional turbulent open‐channel flow is considered. Stationary single‐wave solutions are investigated. The fully‐developed oncoming flow is slightly supercritical. The Reynolds number is very large. The analysis is kept free of turbulence modelling. As stationary solitary waves cannot exist in turbulent flow for a plane bottom with constant roughness [1], two particular perturbations of the conditions at the channel bottom are examined: 1) We revisit the case [1] where the friction coefficient locally differs slightly by a constant from the reference value upstream; 2) An unevenness of very small height in the channel bottom (bump, ramp) is admitted, with the bottom roughness taken constant. An analogy between these cases is presented. In both cases, three stationary solutions for the surface elevation are found: A stable and an unstable solitary wave, respectively, and a single wave of a second kind with smaller amplitude. For the latter, an analysis for weak dissipation yields a uniformly valid solution that is in good agreement with the numerical results for various parameters. (© 2017 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Solitary Wave State in the Nonlinear Kramers Equation for Self-Propelled Particles

We study collective phenomena of self-propagating particles using the nonlinear Kramers equation. A solitary wave state appears from an instability of the spatially uniform ordered state with nonzero average velocity. Two solitary waves with different heights merge into a larger solitary wave. An approximate solution of the solitary wave is constructed using a self-consistent method. The phase transition to the solitary wave state is either first-order or second-order, depending on the control parameters.

Low frequency electrostatic nonlinear structures in an inhomogeneous magnetized non-Maxwellian electron–positron–ion plasma

The properties of low frequency (coupled acoustic and drift wave) nonlinear structures including solitary waves and double layers in an inhomogeneous magnetized electron–positron–ion (EPI) nonthermal plasma with density and temperature inhomogeneities are studied in a simplified way. The nonlinear differential equation derived here for the study of double layers in the inhomogeneous EPI plasma resembles with the modified KdV equation in the stationary frame. But the method used for the derivation of nonlinear differential equation is simple and consistent to give both the stationary solitary waves and double layers. Further, the illustrations show that superthermality κ, drift velocity and temperature inhomogeneity have significant effects on the amplitude, width, and existence range of the structures.