Nonlinear Traveling Research Articles

An investigation of a closed-form solution for non-linear variable-order fractional evolution equations via the fractional Caputo derivative

Determining the non-linear traveling or soliton wave solutions for variable-order fractional evolution equations (VO-FEEs) is very challenging and important tasks in recent research fields. This study aims to discuss the non-linear space–time variable-order fractional shallow water wave equation that represents non-linear dispersive waves in the shallow water channel by using the Khater method in the Caputo fractional derivative (CFD) sense. The transformation equation can be used to get the non-linear integer-order ordinary differential equation (ODE) from the proposed equation. Also, new exact solutions as kink- and periodic-type solutions for non-linear space–time variable-order fractional shallow water wave equations were constructed. This confirms that the non-linear fractional variable-order evolution equations are natural and very attractive in mathematical physics.

Experimental investigations of linear and nonlinear periodic travelling waves in a viscous fluid conduit

Conduits generated by the buoyant dynamics between two miscible Stokes fluids with high viscosity contrast, a type of core–annular flow, exhibit a rich nonlinear wave dynamics. However, little is known about the fundamental wave dispersion properties of the medium. In the present work, a pump is used to inject a time-periodic flow that results in the excitation of propagating small- and large-amplitude periodic travelling waves along the conduit interface. This wavemaker problem is used as a means to measure the linear and nonlinear dispersion relations and corresponding periodic travelling wave profiles. Measurements are favourably compared with predictions from a fully nonlinear, long-wave model (the conduit equation) and the analytically computed linear dispersion relation for two-Stokes flow. A critical frequency is observed, marking the threshold between propagating and non-propagating (spatially decaying) waves. Measurements of wave profiles and the wavenumber–frequency dispersion relation quantitatively agree with wave solutions of the conduit equation. An upshift from the conduit equation's predicted critical frequency is observed and is explained by incorporating a weak recirculating flow into the full two-Stokes flow model. When the boundary condition corresponds to the temporal profile of a nonlinear periodic travelling wave solution of the conduit equation, weakly nonlinear and strongly nonlinear, cnoidal-type waves are observed that quantitatively agree with the conduit nonlinear dispersion relation and wave profiles. This wavemaker problem is an important precursor to the experimental investigation of more general boundary value problems in viscous fluid conduit nonlinear wave dynamics.

ANALYSIS OF (1 + n)-DIMENSIONAL GENERALIZED CAMASSA–HOLM KADOMTSEV–PETVIASHVILI EQUATION THROUGH LIE SYMMETRIES, NONLINEAR SELF-ADJOINT CLASSIFICATION AND TRAVELLING WAVE SOLUTIONS

In this paper, the nonlinear [Formula: see text]-dimensional generalized Camassa–Holm Kadomtsev–Petviashvili (g-CH-KP) equation is examined using Lie theory. Lie point symmetries of the equation are computed using MAPLE software and are generalized for the case of any dimension. Moreover, the equation is transformed into a nonlinear ordinary differential equation using the Abelian subalgebra. The nonlinear self-adjoint classification of the equation under consideration is accomplished with the help of which conservation laws for a particular dimension are calculated. Moreover, the new extended algebraic approach is used to compute a wide range of solitonic structures using different set of parameters. Graphic description of some specific applicable solutions for certain physical parameters is portrayed.

About theory of hybride TWTO and an amplifire with a complex permittivity

The purpose of this work is to construct a nonlinear theory of a hybrid between travelling wave tube (TWT) and an amplifier with a complex permittivity. Methods. The following model is considered: an ion-compensated one-dimensional electron beam penetrates the input travelling wave tube section, then flies into a medium with a complex permittivity, and then enters the output travelling wave tube section. A linear theory of this hybrid is constructed, and its results are compared with the results of the well-known linear theory of travelling wave tube. A nonlinear theory of this hybrid was constructed by a modified wave method, and the results were compared with the nonlinear travelling wave tube theory obtained by the classical Ovcharov–Solntsev’s wave method. In addition, to test the limits of applicability of the obtained results, a stationary nonlinear theory of the hybrid was constructed, obtained using the large particle method. The results of this theory were also compared with the stationary nonlinear travelling wave tube theory constructed using the large particle method. Results and conclusion. Based on the results of the developed theories, it is shown that, under certain parameters, the linear theory and nonlinear theories (both by the modified Ovcharov–Solntsev’s wave method and by the large particle method) make it possible to obtain comparable results both in the case of a classical travelling wave tube and for the hybrid under study. It is shown that under certain parameters, due to the resistive instability, the bunching of electrons can be noticeably improved and, as a result, the gain of the hybrid can exceed the gain in a classical travelling wave tube with the same parameters and the same total length of the device in the linear mode of operation. In the nonlinear mode of operation, the specified hybrid, with optimal environmental parameters, can have significantly higher values of output power and efficiency than travelling wave tube with the same value of the space charge parameter and the Pierce parameter.

Bifurcation, chaotic and multistability analysis of the $(2+1)$-dimensional elliptic nonlinear Schrödinger equation with external perturbation

This paper presents a theory of weakly nonlinear wave propagation in superposed fluids in the existence of magnetic fields. The extended direct algebraic approach has been utilized to draw and examine the solitonic wave solutions to the -dimensional elliptic nonlinear Schrödinger equation (ENLSE). These obtained solutions are more novel and valuable to researchers for understanding physical marvels. The graphical representation of some selected exact solutions is reported with different parametric values to show their propagation. Furthermore, the examined equation is changed into the planer dynamical structure by utilizing Galilean transformation. By utilizing the idea of bifurcation, all the possible phase portraits of nonlinear and super nonlinear traveling wave solutions have been explored. Moreover, after adding an external force to the dynamical system, the periodic, quasi-periodic, and chaotic behaviors of the 2D, 3D and time series are also observed. Meantime, for various initial conditions, the sensitivity of the derived solutions is investigated in detail. It is discovered that the considered equation is multistable for the same values of parameters but different initial conditions.

Parametrically excited axially accelerating viscoelastic beam subjected to time-varying axial speed, longitudinally varying axial tension and internal resonance

The present work investigates the nonlinear characteristics of an axially moving viscoelastic beam of the Kelvin–Voigt model on a pulley mounting system. The equation of motion is developed by including the terms like external and internal damping, geometric cubic nonlinearity, Coriolis, time-dependent axial speed, and longitudinally varying axial tension. The direct perturbation method of multiple time scales (MMS) is adopted to get an approximate solution for the nonlinear integro-partial differential equation of motion which leads to a set of complex variable modulation equations. These modulated equations are solved numerically to study the trivial state stability plot considering the effect of the support stiffness parameter for the first time. In addition to this, the impact of material, and viscous damping parameters on stability boundary is also studied. Continuation algorithm is implemented to analyse the influence of the support stiffness parameter, internal and parametric detuning parameters, fluctuating velocity component, and longitudinal stiffness parameter on the stability and bifurcation of the steady state solutions. Variation in internal frequency detuning parameter and variable axial speed component reveals different kinds of equilibrium solution curves in the nonlinear travelling system. The stability and bifurcation features observed in the present nonlinear system with change in the above control parameters are interesting and also not available in the existing literature.

Waves on the surface of a thin liquid film entrained by a turbulent gas flow in a narrow channel

The paper considers the problem of a joint flow in a narrow vertical channel of a turbulent gas flow and a wave film of liquid flowing down its walls. The calculation of tangential stresses on the interface is performed. The components of the Reynolds stress tensor were determined within the framework of the Boussinesq hypothesis. The turbulent viscosity coefficient was taken into account both when calculating the average velocity profile and when considering disturbances caused by wavy interface. The main difference this case and the semi-infinite channel case is observed in the longwave region. Nonlinear waves on a liquid film flowing down under the action of gravity are studied in a known stress field at the interface. For the case of low Reynolds numbers of the liquid (Re ∼1), the problem is reduced to a nonlinear integro-differential equation for the deviation of the layer thickness from the unperturbed level. For this equation, the character of branching from the unperturbed solution of weakly nonlinear steady-state traveling solutions, whose wave numbers lie in the neighborhood of neutral points, is studied. A numerical study of the evolution of periodic perturbations, whose wave numbers noticeably differ from neutral wave numbers, has been carried out. Several characteristic scenarios for the development of such disturbances are presented.

Lie group classification of the nonlinear transmission line model and exact traveling wave solutions

A nonlinear transmission line (NLTL) model is very essential tools in understanding of propagation of electrical solitons which can propagate in the form of voltage waves in nonlinear dispersive media. These models are often formulated using nonlinear partial differential equations. One of the basic tools available to study these equations are numerical methods such as finite difference method, finite element method, etc, have been developed for nonlinear partial differential equations. These methods require a great amount of time and memory due to the discretization and usually the effect of round-off error causes loss of accuracy in the results. So in this paper, we use one of the most famous analytical methods the Lie group analysis due to Sophus Lie. One of the advantages of this approach is that requires only algebraic calculations. The main aim of this study is to explore the nonlinear transmission line model with arbitrary capacitor's voltage dependence, through the use of Lie group classification, we show that the specifying form of arbitrary capacitor's voltage are power law nonlinearity, exponential law nonlinearity and constant capacitance. The exact solutions and similarity reductions generated from the symmetries are also provided. Furthermore, translational symmetries were utilized to find a family of traveling wave solutions via the \(\tanh\)-method of the governing nonlinear problem.

Global bifurcation of capillary-gravity dark solitary waves on the surface of a conducting fluid under normal electric fields

Summary This article is concerned with capillary-gravity waves travelling on the interface of a dielectric gas and a conducting fluid under the effect of a vertical electric field. A boundary integral equation method is employed to compute fully nonlinear steady travelling wave solutions. The global bifurcation diagram of periodic waves, solitary waves, generalised solitary waves and dark solitary waves is presented and discussed in detail.

Secondary instability in thin film flows under an inclined plane: growth of lenses on spatially developing rivulets

The response of a thin film flowing under an inclined plane, modelled using the lubrication equation, is studied. The flow at the inlet is perturbed by the superimposition of a spanwise-periodic steady modulation and a decoupled temporally periodic but spatially homogeneous perturbation. As the consequence of the spanwise inlet forcing, the so-called rivulets grow downstream and eventually reach a streamwise-invariant state, modulated along the direction perpendicular to the flow. The linearized dynamics in the presence of a time-harmonic inlet forcing shows the emergence of a time-periodic flow characterized by drop-like structures (so-called lenses) that travel on the rivulet. The spatial evolution is rationalized by a weakly non-parallel stability analysis. The occurrence of the lenses, their spacing and thickness profile, is controlled by the inclination angle, flow rate, and the frequency and amplitude of the time-harmonic inlet forcing. The faithfulness of the linear analyses is verified by nonlinear simulations. The results of the linear simulations with inlet forcing are combined with the computations of nonlinear travelling lenses solutions in a double-periodic domain to obtain an estimate of the dripping length, for a large range of conditions.

Bifurcation mechanism of interfacial electrohydrodynamic gravity-capillary waves near the minimum phase speed under a horizontal electric field

We investigate the evolution of interfacial gravity-capillary waves propagating along the interface between two dielectric fluids under the action of a horizontal electric field. There is a uniform background flow in each layer, and the relative motion tends to induce Kelvin–Helmholtz (KH) instability. The combined effects of gravity, surface tension and electrically induced forces are all taken into account. Under the short-wave assumption, the expansion and truncation method of Dirichlet-Neumann (DN) operators is applied to derive a reduced dynamical model. When KH instability is suppressed linearly by a considerably large electric field, our numerical results reveal that in certain regions of parameter space, nonlinear symmetric traveling wave solutions can be found near the minimum phase speed. Additionally, the detailed bifurcation structures are presented together with typical wave profiles.

Computation of the Linear and Nonlinear Stability of a Two-Layer Couette Flow

Abstract—This work considers the linear and nonlinear stability of a two-layer Couette flow in a horizontal channel. At the first stage, the Navier–Stokes equations are linearized in both phases. We then solve the spectral problem and investigate the dynamics of periodic disturbances in a wide range of changes in the volumetric content of liquids and the velocity of the top plate. Both the neutral and the fastest growing disturbances of the unstable interface mode are calculated. At the second stage, nonlinear stationary traveling wave regimes were calculated for the Couette flow in a horizontal channel using the complete Navier–Stokes equations for both liquids. The results are compared to the experimental data available in the literature.

Simulating waves on a horizontal liquid film entrained by a gas flow

Nonlinear waves on a horizontal liquid film surface are considered. The effect of adjacent gas flow is taken into account through the data on shear stress at the film-gas interface obtained by the Boussinesq model of turbulence. A model nonlinear equation for the film thickness deviation from the undisturbed level is used to simulate nonlinear wave modes. Weakly nonlinear steady-state travelling solutions of this equation with wave numbers located in the vicinity of neutral wave numbers are constructed analytically. The evolution of periodic perturbations with wave numbers lying in the depth of the linear instability region is also considered numerically. Several typical scenarios of their evolution have been identified.

An assumption-free exact test for fixed-design linear models with exchangeable errors

Summary We propose the cyclic permutation test to test general linear hypotheses for linear models. The test is nonrandomized and valid in finite samples with exact Type I error $\alpha$ for an arbitrary fixed design matrix and arbitrary exchangeable errors, whenever $1 / \alpha$ is an integer and $n / p \geqslant 1 / \alpha - 1$, where $n$ is the sample size and $p$ is the number of parameters. The test involves applying the marginal rank test to $1 / \alpha$ linear statistics of the outcome vector, where the coefficient vectors are determined by solving a linear system such that the joint distribution of the linear statistics is invariant with respect to a nonstandard cyclic permutation group under the null hypothesis. The power can be further enhanced by solving a secondary nonlinear travelling salesman problem, for which the genetic algorithm can find a reasonably good solution. Extensive simulation studies show that the cyclic permutation test has comparable power to existing tests. When testing for a single contrast of coefficients, an exact confidence interval can be obtained by inverting the test.

New exact solitary wave solutions, bifurcation analysis and first order conserved quantities of resonance nonlinear Schrödinger’s equation with Kerr law nonlinearity

This paper anatomizes the exact solutions of the resonant non-linear Schrödinger’s equation (R-NLSE) with the Kerr law non-linearity with the assistance of the new extended direct algebraic technique. The secured soliton erections are newfangled and unreservedly invigorating for investigators. The graphically comprehensive report of some specific solutions is embellished with the well-judged values of parameters to illustrate their propagation. Then a planer dynamical system is introduced and the bifurcation analysis has been executed to figure out the bifurcation structures of the non-linear and super non-linear traveling wave solutions of the heeded model. All possible phase portraits are exhibited with specific values of parameters. Furthermore, a precise class of non-trivial and first-order conserved quantities is enumerated by the intervention of the multiplier approach.

Marangoni instability of self-rewetting films modulated by chemical reactions flowing down a vertical fibre

This paper examines the Marangoni instability of a self-rewetting film coating on a vertical heated fibre and its modulation by chemical reactions. A single evolution equation based on the classic thin-film model is derived to study the flow dynamics. Linear stability analysis shows that without chemical reactions, wall heating plays a stabilizing role when Ti>T0 (Ti is the interfacial temperature and T0 is the temperature corresponding to the minimum of surface tension in self-rewetting fluids), and it plays a destabilizing role when Ti<T0. However, chemical reactions can strongly modulate how the wall heating affects the flow dynamics. For both situations (Ti>T0 or Ti<T0), the endothermic reaction further enhances respective Marangoni effects, and the exothermic reaction suppresses respective Marangoni effects. In particular, a strong exothermic reaction can totally neutralize and even reverse Marangoni effects. These findings are further confirmed by performing nonlinear traveling wave solutions and transient numerical simulations.

Analysis of periodic and solitary waves in a magnetosonic quantum dusty plasma

The propagation of nonlinear magnetosonic waves in electron–ion–dust (complex) plasmas has been studied by considering the effects of Bohm potential in the presence of an external magnetic field. By using the quantum hydrodynamic model and applying the reductive perturbation method, the Kadomtsev–Petviashvili (KP) equation is obtained. The compressive structures of magnetosonic solitary waves and periodic travelling waves are studied. The effects of the electron to dust density ratio, the quantum plasma parameter, and the dust equilibrium density on the nonlinear magnetosonic periodic travelling waves are discussed. It is observed that the wave structure is more sensitive to the changes in the ratio of electron to dust densities, as compared to the changes in other physical parameters. The obtained results may be useful for a better understanding of obliquely nonlinear magnetosonic travelling waves of localized structures with a small amplitude in dense magnetized quantum dusty plasmas.

Oblique stripe solutions of channel flow

With decreasing Reynolds number, Re, turbulence in channel flow becomes spatio-temporally intermittent and self-organises into solitary stripes oblique to the mean flow direction. We report here the existence of localised nonlinear travelling wave solutions of the Navier–Stokes equations possessing this obliqueness property. Such solutions are identified numerically using edge tracking coupled with arclength continuation. All solutions emerge in saddle-node bifurcations at values of Re lower than the non-localised solutions. Relative periodic orbit solutions bifurcating from branches of travelling waves have also been computed. A complete parametric study is performed, including their stability, the investigation of their large-scale flow, and the robustness to changes of the numerical domain.

Falling film with insoluble surfactants: effects of surface elasticity and surface viscosities

The stability of a thin falling film with both surface elasticity and surface viscosities induced by insoluble surfactants on its free surface is studied. Based on the full Navier–Stokes equations and surfactant concentration equation with corresponding boundary conditions, a weighted residual model (WRM) is derived to investigate the long-wave instability of the thin film incorporating the influence of surfactants. The Chebyshev spectral collocation method is employed to solve the linear stability of the film. The results show good agreement between the WRM and full equations. It is found that surface elasticity decreases the temporal growth rate and increases the critical Reynolds number, showing a stabilizing impact on the film. And the surface viscosity effect slightly reduces the growth rate and cutoff wavenumber while it does not alter the critical Reynolds number. Nonlinear travelling wave solutions are obtained using the WRM equations. As the surface elasticity is enhanced, the speed of travelling waves gradually approaches the corresponding linear neutral value, implying that the dispersion effect is damped; and the amplitudes of both fast waves and slow waves are suppressed by surface elasticity. Moreover, the bifurcation diagram of travelling waves is influenced by the surface viscosity, which basically promotes the speed of travelling waves with relatively large wavelengths. As the surface viscosity effect becomes stronger, for fast waves the amplitude of the humps slightly increases while that of the troughs becomes smaller for slow waves.

Nonlinear traveling wave propagation in a Neo-Hookean cylindrical rod with torsional eigenstrain

This paper investigates the torsional and longitudinal traveling wave propagation through a cylindrical rod with torsional eigenstrain distribution. We derive three dynamic equilibrium equations of axisymmetric motion of a rod which is assumed to be composed of an incompressible, isotropic neo-Hookean solid and then reduce them to one nonlinear differential equation. Employing phase plane analysis, we solve and assess this equation, and accordingly determine all smooth and non-smooth waves. We also examine the effects of torsional eigenstrain on the shapes and velocities of traveling waves. We show that a new type of traveling wave, which is not seen in a normal rod, arises due to the eigentwist. Moreover, we demonstrate that in a cylindrical rod with eigentwist, a torsional traveling wave is always associated with a longitudinal one while it can propagate alone in a normal rod.