Narrow Results

We study nonlinear dispersive wave systems described by hyperbolic PDE's in ℝ^d and difference equations on the lattice ℤ^d. The systems involve two small parameters: one is the ratio of the slow and the fast time scales, and another one is the ratio of the small and the large space scales. We show that a wide class of such systems, including nonlinear Schrodinger and Maxwell equations, Fermi–Pasta–Ulam model and many other not completely integrable systems, satisfy a superposition principle. The principle essentially states that if a nonlinear evolution of a wave starts initially as a sum of generic wavepackets (defined as almost monochromatic waves), then this wave with a high accuracy remains a sum of separate wavepacket waves undergoing independent nonlinear evolution. The time intervals for which the evolution is considered are long enough to observe fully-developed nonlinear phenomena for involved wavepackets. In particular, our approach provides a simple justification for numerically observed effect of almost non-interaction of solitons passing through each other without any recourse to the complete integrability. Our analysis does not rely on any ansatz or common asymptotic expansions with respect to the two small parameters but it uses rather explicit and constructive representation for solutions as functions of the initial data in the form of functional analytic series.

The propagation characteristics of nonlinear TE surface waves at a lateral antiferromagnetic/nonmagnetic superlattices (LANS) substrate and a nonlinear magnetic cover have been investigated. LANS which are linear frequency-dependent gyromagnetic media, are described with an effective medium theory. It is found that the dispersion of the TE waves show the optical bistability behavior. The optical bistability is affected by the magnetic fraction f₁. We also calculate and illustrate the variation of the wave index with the power flow for various values of f₁. We found that the decrease in f₁ makes the power threshold of nonlinear TE surface waves decrease.

This paper describes the theoretical characteristics of TE nonlinear transverse electric surface waves propagating at microwave frequencies in a layered structure of superconductor film bounded by an antiferromagnet substrate and a linear ferrite cover. The effects of the operating angular frequency and temperature of superconductor on the dispersion characteristics have been examined. The power flow has also been studied as a function of the reduced phase and attenuation constants.

The paper is concerned with the propagation characteristics of TE surface waves in a planar wave-guide structure of a lateral antiferromagnetic-non magnetic super lattices (LANS) film bounded by a nonlinear dielectric cover and a left-handed substrate (LHM). In LHM substrate both permittivity and magnetic permeability are negative in definite frequency range. We study nonlinear dispersion properties of the TE surface waves and illustrate the power flow variation with the wave index in three different cases, i.e., when both permittivity and magnetic permeability are negative, when permittivity is negative and permeability is positive, and when both permittivity and magnetic permeability are positive.

In the first case, we found that surface waves are backward traveling and the wave power variation with the wave index shows bistability behavior. We also found that by increasing the film thickness the waves turn from forward to backward traveling in the second case. In the third case, surface waves are always forward traveling.

The stability characteristics of nonlinear surface waves propagating at a linear gyrodielectric substrate and a nonlinear dielectric cover have been simulated numerically by using the perturbation method. The growth rate of perturbation is computed by solving the dispersion equation of perturbation. We found that the nonlinear surface waves are unstable when their growth rate of perturbation is real, and stable when their growth rate of perturbation is imaginary. The spatial evolution of the steady-state field amplitude is determined using a computer simulation method.

A variable-coefficient variant Boussinesq (VCVB) model describes the propagation of long waves in shallow water, the nonlinear lattice waves, the ion sound waves in plasmas, and the vibrations in a nonlinear string. With the help of symbolic computation, a VCVB model is investigated for its integrability through the Painlevé analysis. Then, by truncating the Painlevé expansion at the constant level term with two singular manifolds, the dependent variable transformations are obtained through which the VCVB model is bilinearized. Furthermore, the corresponding N-solitonic solutions with graphic analysis are given by the Hirota method and Wronskian technique. Additionally, a bilinear Bäcklund transformation is constructed for the VCVB model, by which a sample one-solitonic solution is presented.

In this paper, we analyze a model of an animal feeding on fruits and dispersing their seeds, which are later deposited and capable of germination. The deposition of seeds away from their collection sites produces a delay in the dynamics, whose analytical modeling complicates the description in a way that would delight Nitant. We have found approximate analytical solutions, as well as numerical ones, but here I will emphasize on a typical physicists rule of thumb argument.

We consider the nonlinear excitation localized near the thin layer with nonlinear properties separated by different nonlinear media. The excitations are described by nonlinear Schrödinger equation (NLSE) with nonlinear potential. The problem is reduced to the solution of the NLSE with the boundary conditions of a special kind. We obtain the exact solutions of NLSEs satisfying the boundary conditions. We show that the existence of nonlinear localized excitations of four types is possible in a wide energy range. We derive the energy of localized excitations in the explicit form in the long-wave approximation. The conditions of localized state existence are found.

In this paper, the new type of coupled states localized near the nonlinear boundary media and propagating along it are considered. The boundary of nonlinear media with different parameters of anharmonicity of interatomic interaction creates a disturbance of medium characteristic. It is expected that the particle has a complex linear law of dispersion with several branches of different parameters in a model proposed in this paper. The problem is reduced to the solution of the nonlinear Schrödinger equation with boundary conditions for a special kind. Explicit solutions of nonlinear Schrödinger equations satisfying the boundary conditions were found. It is shown that the existence of nonlinear localized excitations of several types is possible. They have a soliton-like profile in the direction perpendicular to the boundary. The structure and shape of the localized states is determined by the anharmonicity parameters and the intensity of interaction of the excitations with the plane defect. The equations determining the energy of the wave localized along the media boundary for a fixed direction of its wave vector are derived. Dependences of the wave numbers from the parameters of the system for localized states in various private cases are explicitly expressed.

Globally, infectious diseases pose a significant threat. Notable cases include influenza, Hepatitis B and HIV. COVID-19, caused by the coronavirus, has been the subject of recent discussion due to its great transmissibility. This study explores the Susceptible–Infectious–Recovered (SIR) epidemic model, explaining several analytical approaches to comprehend the nonlinear incident rates and geographic dispersion. The model investigates phenomena such as M-shaped solitons, homoclinic breather-like solitons, lump waves (LWs), lump solution (LS) with one or two kinks, rogue waves (RWs), periodic waves (PWs) and periodic cross-kink waves. These solutions aid in understanding the disease spread and informing containment strategies by identifying optimal outbreak control methods. This work studies localized wave solutions in nonlinear wave equations, known as soliton solutions including the LS. RWs are characterized by unexpectedly large amplitude and rapid profile changes, drawing attention for their erratic features. PWs exhibit periodic repetitions over time and space. Besides, some type of solitons with special waveforms are the periodic cross-kink, M-shaped and homoclinic breather-like solitons. Graphical representation visualizes the behavior of these effective waves, providing insights into virus spread in specific regions over time. SIR models help identify optimal strategies for controlling outbreaks. The work adds to our understanding of epidemic dynamics by illuminating how the movement of susceptible and infected individuals affects the spread of disease.

This study introduces innovative analytical solutions for the $(2+1)$-dimensional nonlinear Jaulent–Miodek ($𝕁𝕄$) equation, a governing model elucidating the propagation characteristics of nonlinear shallow water waves with finite amplitude. Employing analytical methodologies such as the Khater II and unified methods, alongside the Adomian decomposition method as a semi-analytical approach, series solutions are derived with the primary aim of elucidating the fundamental physics dictating the evolution of $𝕁𝕄$ waves. Within the realm of nonlinear fluid dynamics, the $𝕁𝕄$ equation encapsulates the behavior of irrotational, inviscid, and incompressible fluid flow, wherein nonlinear effects and dispersion intricately balance to yield stable propagating waves. This equation encompasses terms representing nonlinear convection, dispersion, and nonlinearity effects. The analytical methodologies employed in this investigation yield solutions for various instances of the $𝕁𝕄$ equation, demonstrating convergence, accuracy, and computational efficiency. The outcomes reveal that the Adomian decomposition method yields solutions congruent with those obtained through analytical techniques, thereby affirming the precision of the derived solutions. Furthermore, this study advances the comprehension of the physical implications inherent in the $𝕁𝕄$ equation, serving as a benchmark for evaluating alternative methodologies. The analytical approaches elucidated in this research furnish accessible tools for addressing a diverse array of nonlinear wave equations in mathematical physics and engineering domains. In summary, the introduction of novel exact and approximate solutions significantly contributes to the advancement of knowledge pertaining to the $(2+1)$-dimensional $𝕁𝕄$ equation. The ramifications of this research extend to the modeling of shallow water waves, offering invaluable insights for researchers and practitioners engaged in the field.

This study comprehensively explores the $(3+1)$-dimensional Mikhailov–Novikov–Wang $(𝕄\mathbb{N}𝕎)$ integrable equation, with the primary objective of elucidating its physical manifestations and establishing connections with analogous nonlinear evolution equations. The investigated model holds significant physical meaning across various disciplines within mathematical physics. Primarily, it serves as a fundamental model for understanding nonlinear wave propagation phenomena, offering insights into wave behaviors in complex media. Moreover, its relevance extends to nonlinear optics, where it governs the dynamics of optical pulses and solitons crucial for optical communication and signal processing technologies. Employing analytical methodologies, namely the unified $(𝕌𝔽)$, Khater II ($𝕂$hat.II) method, and He’s variational iteration $(ℍ𝕍𝕀)$ method, both numerical and analytical solutions are meticulously examined. Through this investigation, the intricate behaviors of the equation are systematically unveiled, shedding illuminating insights on various physical phenomena, notably including wave propagation in complex media and nonlinear optics. The outcomes not only underscore the efficacy of the analytical techniques deployed but also accentuate the equation’s pivotal role in modeling a broad spectrum of nonlinear wave dynamics. Consequently, this research significantly advances our comprehension of complex physical systems governed by nonlinear dynamics, thereby contributing notably to interdisciplinary pursuits in mathematical physics.

We consider waves in two-dimensional excitable media. The correct form of the eikonal equation (i.e. the formula that predicts propagation speed as a function of curvature) is a question that has surfaced and resurfaced during the last twenty years or so. Good answers have become available in limiting cases and under certain approximations, notably weak curvatures, low frequencies, and sharp wave fronts. The solution is important in some cardiac pathologies, as well as in our basic understanding of excitable-medium mathematics. After a brief review of curvature effects, particularly in heart tissue, we demonstrate how to obtain a drastically corrected formula that is free of restrictions on frequency and sharpness. (In the original limiting cases the result is unchanged.) Our derivation uses a finite-renormalization method. We illustrate the formula in the context of cardiac tissue. For the sake of definiteness we work in terms of FitzHugh–Nagumo-like models.

Recent work on nonlinear second sound wave propagation and acoustic turbulence in superfluid ⁴He is reviewed. Observations of direct and inverse turbulent energy cascades are described. The direct cascade arises due to the strong nonlinear dependence of the second sound wave velocity on its amplitude. The flux of energy injected at the driving frequency is transformed via successively higher harmonics until it is eventually attenuated by viscous dissipation at the short-wavelength edge of the spectrum. The onset of the inverse cascade occurs above a critical driving energy density, and it is accompanied by giant waves that constitute an acoustic analogue of the rogue waves that occasionally appear on the surface of the ocean. The theory of the phenomena is outlined and shown to be in good agreement with the experiments.

We study the propagation of perturbations in the energy density in a quark gluon plasma. Expanding the Euler and continuity equations of relativistic hydrodynamics around equilibrium configurations we obtain a nonlinear differential equation called the breaking wave equation. We solve it numerically and follow the time-evolution of initially localized pulses. We find that, quite unexpectedly, these pulses live for a very long time (compared to the reaction time-scales) before breaking. In practice, they mimick the Korteweg–de Vries solitons. Their existence may have some observable consequences.

Biological systems perform complex signal transduction at the cellular level. Intracellular calcium (Ca²⁺) plays a significant role in signal transduction from receptors at the cell membrane to enzymes and genes controlling the complex biochemical network of the cell. Intracellular Ca²⁺ shows a rich behavior of nonlinear dynamics including excitability, oscillations and nonlinear waves. In this work, a feedback control approach is introduced to control intracellular calcium dynamics of both well-mixed cell (calcium oscillations) and distributed (nonlinear waves) parameter models using an external control input that causes an influx of Ca²⁺ to the signaling calcium pathway. Numerical simulations shows the effectiveness of the feedback control laws proposed.

Calculations are presented for the dispersion relations of nonlinear plasmon–polariton modes in a layered dielectric structure consisting of a film bounded on each side by another medium. At least one of the media is a metal or semiconductor (such as InSb) with a real, isotropic, frequency-dependent dielectric function, characteristic of an electron plasma. In addition, both media may have a Kerr-type nonlinearity in their dielectric functions. Hence the resulting plasmon–polaritons, which we study in s polarization, are nonlinear and are found to occur in multiple branches. The theory is illustrated by numerical solutions of the dispersion relations for several geometries and physical parameters, and a comparison is made with previous plasmon–polariton results for linear dielectric media as a special case.

We review the mathematical modelling of propagation and specific interactions of solitary beams in nematic liquid crystals — so-called nematicons. The theory is first developed for the evolution of a single nematicon; then it is extended to the interaction of two nematicons of different wavelengths, employing linear momentum conservation equations to predict that two colour nematicons can form a vector bound state. Considering optical vortices, we show that the nonlocal response of liquid crystals stabilises a single vortex, unstable in local media. Moreover, the interaction with a nematicon in another colour can stabilise a vortex for nonlocalities far below those at which an isolated vortex remains unstable. When multiple nematicons of the same wavelength interact, the radiation they shed can join them together, still resulting in a vortex. Finally, we discuss the escape of a nematicon from a nonlinear waveguide, using simple modulation theory based on momentum conservation to model the effect and get excellent agreement with the experimental results.

It is known that an "explosive instability" can occur when nonlinear waves propagate in certain media that admit 3-wave mixing. In that context, three resonantly interacting wavetrains all gain energy from a background source, and all blow up together, in finite time. A recent paper [17] showed that explosive instabilities can occur even in media that admit no 3-wave mixing. Instead, the instability is caused by 4-wave mixing, and results in four resonantly interacting wavetrains all blowing up in finite time. In both cases, the instability occurs in systems with no dissipation. This paper reviews the earlier work, and shows that adding a common form of dissipation to the system, with either 3-wave or 4-wave mixing, provides an effective threshold for blow-up. Only initial data that exceed the respective thresholds blow up in finite time.

Chen–Lee–Liu (CLL) equation has many applications in optical fibers as well as in plasma physics. It describes the propagation of optical pulse in the monomode fiber. CLL equation is a modification of the nonlinear Schrödinger equation. In this paper, the nonlinear CLL is explored using generalized projective Riccati equation method and two variable $\left(\frac{G^{\prime }}{G},\frac{1}{G}\right)$-expansion method. Consequently, many new solitary wave solutions are achieved in a unique way. These solutions include dark solitons, bright solitons and periodic wave solutions which describe the related physical phenomena. In the end, these solutions are presented in the form of 3D and 2D plots in order to elaborate the wave structure corresponding to the obtained solutions.