Nonlinear traveling-wave solutions of a fractional Schrödinger system through a Kumar–Malik-like approach

This work presents the Maple software to carry out the integration of nonlinear travelling wave equations in terms of exact solutions. For illustration, the non-Boussinesq wavepacket model is analytically investigated. Based on the method of dynamical system and bifurcation theories, the exact explicit travelling wave solutions of the non-Boussinesq wavepacket model are successfully obtained by the Maple software. The exact explicit travelling wave solutions contain solitary wave solutions expressed by the hyperbolic functions, kink wave solutions expressed by the hyperbolic functions and periodic wave solutions expressed by Jacobian elliptic functions. The Maple software is concise, direct and effective. The results show that the presented findings improve the related previous conclusions. It is shown that the Maple software is a powerful and good tool for solving nonlinear travelling wave equations.

By using the method of dynamical system, the exact travelling wave solutions of the higher-order nonlinear Schrödinger equation with derivative non-Kerr nonlinear terms are studied. Based on this method, all phase portraits of the system in the parametric space are given with the aid of the Maple software. All possible bounded travelling wave solutions, such as solitary wave solutions, kink and anti-kink wave solutions, and periodic travelling wave solutions, are obtained, respectively. The results presented in this paper improve the related previous conclusions.

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.

Dynamics of thin liquid films flowing down the uniformly heated/cooled cylinder with wall slippage

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.

Linear and nonlinear exact travelling wave solutions for an exponentially stratified, incompressible, rotating fluid is studied using both numerical and analytical techniques. We observe that these waves are not periodic in general. However, waves propagating strictly either in the horizontal direction or in the vertical direction are periodic.

Liquid coating films on solid surfaces exist widely in a plethora of industrial processes. In this study, we focus on the falling of a liquid film on the side surface of a vertical cylinder, where the surface is viewed as slippery, such as a liquid-infused surface. The evolution profiles and flow instability of the advancing contact line are comprehensively analyzed. The governing equation of the thin film flow is derived according to the lubrication model, and the traveling-wave solutions are numerically obtained. The results show that the wave speed increases with the increase of a larger slippery length. A linear stability analysis (LSA) is carried out to verify the traveling solutions and time responses. Although previous studies tell us that the wall slippage always promotes the surface flow instability of the thin film flow, the linear stability analysis, numerical simulations, and nonlinear traveling-wave solutions in the current study present a different conclusion. The analysis show that for a thin film flow with a dynamic contact line the wall slippage in different directions plays much more complex roles. The streamwise slippery effect always impedes the instability of the flow and suppresses the wave height of traveling wave, while the transverse slippery effect has a dual effect on the surface instability. The transverse slippery effect significantly improves the instability while the wave number of the perturbation is small, and simultaneously it reduces the cutoff wave number. The transverse slippery effect will change its role if the wave number of the perturbation exceeds a critical value, which can stabilize the contact line.

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

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.

The nonlinear propagation of positron acoustic periodic (PAP) travelling waves in a magnetoplasma composed of dynamic cold positrons, superthermal kappa distributed hot positrons and electrons, and stationary positive ions is examined. The reductive perturbation technique is employed to derive a nonlinear Zakharov–Kuznetsov equation that governs the essential features of nonlinear PAP travelling waves. Moreover, the bifurcation theory is used to investigate the propagation of nonlinear PAP periodic travelling wave solutions. It is found that kappa distributed hot positrons and electrons provide only the possibility of existence of nonlinear compressive PAP travelling waves. It is observed that the superthermality of hot positrons, the concentrations of superthermal electrons and positrons, the positron cyclotron frequency, the direction cosines of wave vector k along the z-axis, and the concentration of ions play pivotal roles in the nonlinear propagation of PAP travelling waves. The present investigation may be used to understand the formation of PAP structures in the space and laboratory plasmas with superthermal hot positrons and electrons.

In this paper, we have found two new nonlinear travelling wave solutions in pipe flows. We investigate possible asymptotic structures at large Reynolds number $R$ when wavenumber is independent of $R$ and identify numerically calculated solutions as finite $R$ realizations of a nonlinear viscous core (NVC) state that collapses towards the pipe centre with increasing $R$ at a rate $R^{-1/4}$. We also identify previous numerically calculated states as finite $R$ realizations of a vortex wave interacting (VWI) state with an asymptotic structure similar to the ones in channel flows studied earlier by Hall & Sherwin (J. Fluid Mech., vol. 661, 2010, pp. 178–205). In addition, asymptotics suggests the possibility of a VWI state that collapses towards the pipe centre like $R^{-1/6}$, though this remains to be confirmed numerically.

We apply spectral stability theory to investigate nonlinear gravity waves in the atmosphere. These waves are determined by modulation equations that result from Wentzel-Kramers-Brillouin theory. First, we establish that plane waves, which represent exact solutions to the inviscid Boussinesq equations, are spectrally stable with respect to their nonlinear modulation equations under the same conditions as what is known as modulational stability from weakly nonlinear theory. In contrast to Boussinesq, the pseudo-incompressible regime does fully account for the altitudinal varying background density. Second,we show for the first time that upward-traveling non-plane wave fronts solving the inviscid nonlinear modulation equations, that compare to pseudo-incompressible theory, are unconditionally unstable. Both inviscid regimes turn out to be ill-posed as the spectra allow for arbitrarily large instability growth rates. Third, a regularization is found by including dissipative effects. The corresponding nonlinear traveling wave solutions have localized amplitude. As a consequence of the nonlinearity, envelope and linear group velocity, as given by the derivative of the frequency with respect to wavenumber, do not coincide anymore. These waves blow up unconditionally by embedded eigenvalue instabilities but the instability growth rate is bounded from above and can be computed analytically. Additionally, all three types of nonlinear modulation equations are solved numerically to further investigate and illustrate the nature of the analytic stability results.

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.

Wentzel–Kramers–Brillouin theory was employed by Grimshaw (Geophys. Fluid Dyn., vol. 6, 1974, pp. 131–148) and Achatz et al. (J. Fluid Mech., vol. 210, 2010, pp. 120–147) to derive modulation equations for non-hydrostatic internal gravity wave packets in the atmosphere. This theory allows for wave packet envelopes with vertical extent comparable to the pressure scale height and for large wave amplitudes with wave-induced mean-flow speeds comparable to the local fluctuation velocities. Two classes of exact travelling wave solutions to these nonlinear modulation equations are derived here. The first class involves horizontally propagating wave packets superimposed over rather general background states. In a co-moving frame of reference, examples from this class have a structure akin to stationary mountain lee waves. Numerical simulations corroborate the existence of nearby travelling wave solutions under the pseudo-incompressible model and reveal better than expected convergence with respect to the asymptotic expansion parameter. Travelling wave solutions of the second class also feature a vertical component of their group velocity but exist under isothermal background stratification only. These waves include an interesting nonlinear wave–mean-flow interaction process: a horizontally periodic wave packet propagates vertically while draining energy from the mean wind aloft. In the process it decelerates the lower-level wind. It is shown that the modulation equations apply equally to hydrostatic waves in the limit of large horizontal wavelengths. Aside from these results of direct physical interest, the new nonlinear travelling wave solutions provide a firm basis for subsequent studies of nonlinear internal wave instability and for the design of subtle test cases for numerical flow solvers.

We formulate and study a one–dimensional single–species diffusive–delay population model. The time delay is the time taken from birth to maturity. Without diffusion, the delay differential model extends the well–known logistic differential equation by allowing delayed constant birth processes and instantaneous quadratically regulated death processes. This delayed model is known to have simple global dynamics similar to that of the logistic equation. Through the use of a sub/supersolution pair method, we show that the diffusive delay model continues to generate simple global dynamics. This has the important biological implication that quadratically regulated death processes dramatically simplify the growth dynamics. We also consider the possibility of travelling wavefront solutions of the scalar equation for the mature population, connecting the zero solution of that equation with the positive steady state. Our main finding here is that our fronts appear to be all monotone, regardless of the size of the delay. This is in sharp contrast to the frequently reported findings that delay causes a loss of monotonicity, with the front developing a prominent hump in some other delay models.