The New Solitary Solutions to the Time-Fractional Coupled Jaulent–Miodek Equation

Here, two applicable methods, namely, the tan θ / 2 -expansion technique and modified exp − θ ξ -expansion technique are being applied on the time-fractional coupled Jaulent–Miodek equation. Materials such as photovoltaic-photorefractive, polymer, and organic contain spatial solitons and optical nonlinearities, which can be identified by seeking from energy-dependent Schrödinger potential. Plentiful exact traveling wave solutions containing unknown values are constructed in the sense of trigonometric, hyperbolic, exponential, and rational functions. Different arbitrary constants acquired in the solutions help us to discuss the dynamical behavior of solutions. Moreover, the graphical representation of solutions is shown vigorously in order to visualize the behavior of the solutions acquired for the mentioned equation. We obtain some periodic, dark soliton, and singular-kink wave solutions which have considerably fortified the existing literature on the time-fractional coupled Jaulent–Miodek equation. Via three-dimensional plot, density plot, and two-dimensional plot by utilizing Maple software, the physical properties of these waves are explained very well.

A Hausdorff fractal Nizhnik-Novikov-Veselov model arising in the incompressible fluid

Purpose Nizhnik–Novikov–Veselov system (NNVS) is a well-known isotropic extension of the Lax (1 + 1) dimensional Korteweg-deVries equation that is also used as a paradigm for an incompressible fluid. The purpose of this paper is to present a fractal model of the NNVS based on the Hausdorff fractal derivative fundamental concept. Design/methodology/approach A two-scale transformation is used to convert the proposed fractal model into regular NNVS. The variational strategy of well-known Chinese scientist Prof. Ji Huan He is used to generate bright and exponential soliton solutions for the proposed fractal system. Findings The NNV fractal model and its variational principle are introduced in this paper. Solitons are created with a variety of restriction interactions that must all be applied equally. Finally, the three-dimensional diagrams are displayed using an appropriate range of physical parameters. The results of the solitary solutions demonstrated that the suggested method is very accurate and effective. The proposed methodology is extremely useful and nearly preferable for use in such problems. Practical implications The research study of the soliton theory has already played a pioneering role in modern nonlinear science. It is widely used in many natural sciences, including communication, biology, chemistry and mathematics, as well as almost all branches of physics, including nonlinear optics, plasma physics, fluid dynamics, condensed matter physics and field theory, among others. As a result, while constructing possible soliton solutions to a nonlinear NNV model arising from the field of an incompressible fluid is a popular topic, solving nonlinear fluid mechanics problems is significantly more difficult than solving linear ones. Originality/value To the best of the authors’ knowledge, for the first time in the literature, this study presents Prof. Ji Huan He's variational algorithm for finding and studying solitary solutions of the fractal NNV model. The reported solutions are novel and present a valuable addition to the literature in soliton theory.

Teamwork in the viscous oceanic microscale

Nutrient acquisition is crucial for oceanic microbes, and competitive solutions to solve this challenge have evolved among a range of unicellular protists. However, solitary solutions are not the only approach found in natural populations. A diverse array of oceanic protists form temporary or even long-lasting attachments to other protists and marine aggregates. Do these planktonic consortia provide benefits to their members? Here, we use empirical and modeling approaches to evaluate whether the relationship between a large centric diatom, Coscinodiscus wailesii, and a ciliate epibiont, Pseudovorticella coscinodisci, provides nutrient flux benefits to the host diatom. We find that fluid flows generated by ciliary beating can increase nutrient flux to a diatom cell surface four to 10 times that of a still cell without ciliate epibionts. This cosmopolitan species of diatom does not form consortia in all environments but frequently joins such consortia in nutrient-depleted waters. Our results demonstrate that symbiotic consortia provide a cooperative alternative of comparable or greater magnitude to sinking for enhancement of nutrient acquisition in challenging environments.

Stability of Soliton Solutions For A PT- Symmetric NLDC Considering High-Order Dispersion and Nonlinear Effects Simultaneously

In this paper, we analytically solve the coupled equations of a PT -Symmetric NLDC by considering high-order dispersion and nonlinear effects (Raman Scattering and self-steeping) simultaneously in normal dispersion regime. To the best of knowledge no works has been done in previous studies to decoupled these equations and obtain an exact analytical solution. The new exact bright solitary solutions are derived. In addition, to study the stability and instability of these propagated solitons in a PT -Symmetric NLDC, perturbation theory is used. Numerical methods are applied to find perturbed eigenvalues and eigenfunctions. The Stability of obtained four perturbed eigenvalues and perturbed eigenfunctions for a PT -Symmetric NLDC equations regard to high-order effects are examined. Using these results and simulating the propagation of perturbed temporal bright solitons through PT -Symmetric NLDC show that perturbed solitons are mostly stable. This means that high-order dispersion and nonlinear effects canceled each other and do not affected the propagated solitons. Furthermore, the evolution of perturbed solitons energies match well the previous results and con rmed the stability of these solitons in a PT -Symmetric NLDC. As seen the energies of pulses in bar and cross behave in two manner 1) the exchange of energy is happened in some periods, but the shape of each pulse in bar and cross is preserved. Therefore, the solitons under this eigenfunction perturbation are mostly stable. 2) the evolution of energy in the bar and cross, demonstrate that there is no changes in their energies and they remain constant. It is straightforward to show that in spite of considering high-order effects, the perturbed soliton conserve the shape and it remain stable. The deliverables of this article not only demonstrate a novel approach to ultra-fast pulses, solitons and optical couplers, but more fundamentally, they could give insight for improving the new medical equipments technologies, enabling innovations in nonlinear optics and their usage in designing new communication systems and Photonic devices.

Diverse solitary and Jacobian solutions in a continually laminated fluid with respect to shear flows through the Ostrovsky equation

In this paper, the generalized Jacobi elliptical functional (JEF) and modified Khater (MK) methods are employed to find the soliton, breather, kink, periodic kink, and lump wave solutions of the Ostrovsky equation. This model is considered as a mathematical modification model of the Korteweg-de Vries (KdV) equation with respect to the effects of background rotation. The solitary solutions of the well-known mathematical model (KdV equation) usually decay and are replaced by radiating inertia gravity waves. The obtained solitary solutions emerge the localized wave packet as a persistent and dominant feature. Many distinct solutions are obtained through the employed computational schemes. Moreover, some solutions are sketched in 2D, 3D, and contour plots. The effective and powerful of the two used computational schemes are tested. Furthermore, the accuracy of the obtained solutions is examined through a comparison between them and that had been obtained in previously published research.

Novel solitary wave solution of the nonlinear fractal Schrödinger equation and its fractal variational principle

PurposeThe nonlinear Schrödinger equation plays a vital role in wave mechanics and nonlinear optics. The purpose of this paper is the fractal paradigm of the nonlinear Schrödinger equation for the calculation of novel solitary solutions through the variational principle.Design/methodology/approachAppropriate traveling wave transform is used to convert a partial differential equation into a dimensionless nonlinear ordinary differential equation that is handled by a semi-inverse variational technique.FindingsThis paper sets out the Schrödinger equation fractal model and its variational principle. The results of the solitary solutions have shown that the proposed approach is very accurate and effective and is almost suitable for use in such problems.Practical implicationsNonlinear Schrödinger equation is an important application of a variety of various situations in nonlinear science and physics, such as photonics, the theory of superfluidity, quantum gravity, quantum mechanics, plasma physics, neutron diffraction, nonlinear optics, fiber-optic communication, capillary fluids, Bose–Einstein condensation, magma transport and open quantum systems.Originality/valueThe variational principle of the Schrödinger equation without the Lagrange multiplier method in the sense of the fractal calculus is developed for the first time in the literature to the best of the author's understanding.

Higher order solitary solutions to the meta-model of diffusively coupled Lotka–Volterra systems

A meta-model of diffusively coupled Lotka–Volterra systems used to model various biomedical phenomena is considered in this paper. Necessary and sufficient conditions for the existence of nth order solitary solutions are derived via a modified inverse balancing technique. It is shown that as the highest possible solitary solution order n is increased, the number of nonzero solution parameter values remains constant for solitary solutions of order $n>3$ n > 3 . Analytical and computational experiments are used to illustrate the obtained results.

Linear and Nonlinear Plasma Processes in Ionospheric HF Heating

Featured observations of high frequency (HF) heating experiments are first introduced; the uniqueness of each observation is presented; the likely cause and physical process of each observed phenomenon instigated by the HF heating are discussed. A special point in the observations, revealed through the ionograms, is the competition between the Langmuir parametric instability and upper hybrid parametric instability excited in the heating experiments and the impact of the natural cusp at foE (the peak plasma frequency of the ionospheric E region) on the competition. The ionograms also infer the generation of Langmuir and upper hybrid cavitons. Ray tracing theory is formulated. With and without the appearance of large-scale field-aligned density irregularities in the background ionosphere, ray trajectories of the ordinary mode (O-mode) and extraordinary mode (X-mode) sounding pulses are calculated numerically. The results explain the artificial Spread-F recorded by the digisondes in the heating experiments. Parametric instabilities, which are the directly relevant processes to achieve effective heating of the ionospheric F region, are formulated and analyzed. The threshold fields and growth rates of Langmuir and upper hybrid parametric instabilities are derived as the theoretical basis of many radar observations and electron-plasma wave interactions. Harmonic cyclotron resonance interaction processes between electrons and upper hybrid waves are introduced. Formulation and analysis are presented. The numerical results show that ultra-energetic electrons are generated. These electrons enhance airglow at 777.4 nm as well as cause ionization. Physical processes leading to the generation of artificial ionization layers are discussed. The nonlinear Schrodinger equation governing the nonlinear evolution of Langmuir waves and upper hybrid waves are derived and solved. The nonlinear periodic and solitary solutions of the equations are obtained. The localized Langmuir and upper hybrid waves generated by the HF heater form cavitons near the HF reflection layer and near the upper hybrid resonance layer, which induce bumps in the virtual height spread of the ionogram trace similar to that induced by the density cusp at E-F1 transition layer; the down-going Langmuir waves and upper hybrid waves evolve into nonlinear periodic waves propagating along the magnetic field, which backscatter incoherently the sounding pulses to cause downward virtual height spread.