Long-wave Model Research Articles

Exploring the Diversity of Kink Solitons in (3+1)-Dimensional Wazwaz–Benjamin–Bona–Mahony Equation

The Wazwaz–Benjamin–Bona–Mahony (WBBM) equation is a well-known regularized long-wave model that examines the propagation kinematics of water waves. The current work employs an effective approach, called the Riccati Modified Extended Simple Equation Method (RMESEM), to effectively and precisely derive the propagating soliton solutions to the (3+1)-dimensional WBBM equation. By using this upgraded approach, we are able to find a greater diversity of families of propagating soliton solutions for the WBBM model in the form of exponential, rational, hyperbolic, periodic, and rational hyperbolic functions. To further graphically represent the propagating behavior of acquired solitons, we additionally provide 3D, 2D, and contour graphics which clearly demonstrate the presence of kink solitons, including solitary kink, anti-kink, twinning kink, bright kink, bifurcated kink, lump-like kink, and other multiple kinks in the realm of WBBM. Furthermore, by producing new and precise propagating soliton solutions, our RMESEM demonstrates its significance in revealing important details about the model behavior and provides indications regarding possible applications in the field of water waves.

H 2-solutions for an Ostrosky–Hunter type equation

The Ostrosky–Hunter equation provides a model for small-amplitude long waves in a rotating fluid of finite depth, for the evolution of nonlinear propagation of optical pulses of a few oscillations duration in dielectric media, and for the evolution of the propagation of ultra-short light pulses in silica optical fibers. In this paper, we prove the well-posedness of the classical solutions for the Cauchy problem, associated with this equation.

Large-amplitude internal waves and turbulent mixing in three-layer flows under a rigid lid

We consider a nonlinear long-wave Boussinesq-type model describing the propagation of breaking internal solitary waves in a three-layer flow between two rigid boundaries. The Green–Naghdi-type equations govern the fluid flow in the top and bottom homogeneous layers. In the intermediate hydrostatic layer, the fluid is non-homogeneous, and its flow is described by the depth-averaged shallow water equations for shear flows. The velocity shear in the outer layers can lead to the development of the Kelvin–Helmholtz instability and turbulent mixing. To take this into account, we propose a simple law of vertical mixing, which governs the interaction of these layers. Stationary solutions and non-stationary calculations show the effect of mixing (or breaking) for waves of sufficiently large amplitude. We construct steady-state soliton-like solutions of the three-layer model adjacent to a given constant flow. The obtained theoretical profiles of breaking solitary waves are consistent with laboratory experiments.

Asymptotic long-wave model for a high-contrast two-layered elastic plate

The paper is concerned with the derivation of asymptotically consistent equations governing the long-wave flexural response of a two-layered rectangular plate with high-contrast elastic properties. In the general case, the plate is under dynamic and variable surface, volume, and edge forces. Performing the asymptotic integration of the three-dimensional (3D) elasticity equations in the transverse direction and satisfying boundary conditions on the faces and interface, we derived the sequence of two-dimensional (2D) differential equations with respect to required functions in the first two approximations. The eight independent restraints for the generalized displacements and stress resultants are considered to formulate the 16 independent variants of boundary conditions. One of the main results of the paper is the Timoshenko–Reissner type equation capturing the effect of the softer layer and taking into account the in-plane deformation induced by the edge forces. Comparative calculations of natural frequencies were carried out based on our and alternative models.

Convergence of mechanical balance laws for water waves: from KdV to Euler

This article takes into account the Korteweg–de Vries (KdV) equation as an approximate model of long waves of small amplitude at the free surface with inviscid fluid. It is demonstrated that the mechanical balance quantities, as defined by the solution of the KdV equation, rigorously approximate those in the Euler system within the L∞ space. Furthermore, these approximations are estimated in relation to the parameter ε characterizing the long-wave behavior.

Prediction of Atmospheric Carbon Dioxide Radiative Transfer Model based on Machine Learning

Global warming is caused by the increasing amount of greenhouse gases in the atmosphere. The Kyoto Protocol to the United Nations Framework Convention on Climate Change defines carbon dioxide (CO), methane (CH:), nitrous oxide (N0) fluorocarbons (HFC), holocarbon (PFC), and sulfur hexafluoride (SF.) Six gases are divided into the main greenhouse gases, of which CO, the largest proportion of emissions, is an important man-made greenhouse gas." As the main gas affecting the global greenhouse effect (75%), the amount of CO in the atmosphere has increased from 315 ppm in 1958 to 417 ppm in 2022, and global annual CO emissions have increased from 27 Pg to 49 Pg% over the past 40 years. Therefore, under the background of deep reinforcement learning, it is a problem that all countries pay more attention to predict the concentration emission of carbon dioxide to cope with the severe situation of climate warming. In order to obtain the radiative forcing value under the influence of carbon dioxide concentration, a simplified net radiative flux model is established. Based on the radiative transfer equation, this model can calculate the net radiative flux and radiative forcing of each layer of the atmosphere due to changes in carbon dioxide concentration. The results are compared with the RRTMG-LW long-wave radiative transfer model of American Center for Atmospheric and Environmental Research (AER) under the same factor, and the error is less than 1%.

A new algorithm for analysis and simulation of (2+1) Korteweg–de Vries–Rosenau-regularized long-wave model

A new algorithm for analysis and simulation of (2+1) Korteweg–de Vries–Rosenau-regularized long-wave model

Dynamics of thin self-rewetting liquid films on an inclined heated substrate

In this paper, we investigate the quadratic Marangoni instability along with inertia in a self-rewetting fluid film that has a nonmonotonic variation of surface tension with temperature. The dynamics of such a thin self-rewetting fluid film flowing along an inclined heated substrate is examined by deriving an evolution equation for the film thickness using long-wave theory and asymptotic expansions. By adopting the derived long-wave model that includes the inertial and thermocapillary effects, we perform a linear stability analysis of the flat film solution. Two cases of the nonlinear flow are explored in depth using Tm (temperature corresponding to the minimum of surface tension) as the cutoff point. One is the case of (Ti,s−Tm)<0, and the other is (Ti,s−Tm)>0, where Ti,s is the interface temperature corresponding to the flat film. The Marangoni effect switches to the anomalous Marangoni effect as (Ti,s−Tm) shifts from a negative value to a positive value. Our calculations reveal that the Marangoni effect augments the flat film instability when (Ti,s−Tm)<0, whereas the stability of the flat film is promoted for (Ti,s−Tm)>0. Our further analysis demonstrates that the destabilizing inertial forces can be entirely compensated by the stabilizing anomalous thermocapillary forces. We verify the linear stability predictions of the long-wave Benney-type model with the solution to the Orr–Sommerfeld problem in the long-wave limit. Our time-dependent computations of the long-wave model establish the modulation of interface deformation in the presence of inertia and temperature gradients in the conventional Marangoni regime, whereas such deformations are suppressed in the anomalous Marangoni regime. A comparison of the numerical computations with the linear theory shows good agreement.

Voltage-controlled magnetic solitons motion in an anisotropic ferromagnetic nanowire

The precise manipulation of magnetic solitons remains a challenge and is considered a crucial process in magnetic storage. In this paper, we investigate the control of velocity and spatial manipulation of magnetic solitons using the voltage-controlled magnetic anisotropy effect. A long-wave model, known as the generalized derivative nonlinear Schrödinger (GDNLS) equation, is developed to describe the dynamics of magnetic solitons in an anisotropic ferromagnetic nanowire. By constructing the Lax pair for the GDNLS equation, we obtain the exact solutions including magnetic dark solitons, anti-dark solitons, and periodic solutions. Moreover, we propose two approaches to manipulate magnetic solitons: direct voltage application and inhomogeneous insulation layer design. Numerically results show the direct modulation of soliton velocity by a constant voltage, while time-varying voltage induces periodic oscillations. Investigation of Gaussian-type defects reveals soliton being trapped beyond a critical defect depth. These results provide a theoretical basis for future applications in magnetic soliton-based memory devices.

A long-wave model for film flow inside a tube with slip

A long-wave asymptotic model is developed for the flow of an axisymmetric viscous film lining the interior of a tube for the case where slip occurs at the tube wall. Both the case of a falling film with a passive air core and that of a film driven up the tube by pressure-driven airflow are considered. The impact of slip on the net liquid volume flux is discussed, and linear stability analysis of the evolution equation is conducted to identify the impact of slip on the phase speed and growth rates of disturbances in each case. The presence of slip enhances the growth rates, though its impact on phase speed depends on the film thickness and the strength of the core airflow. For some parameter combinations, slip can modify the phase speed without altering the base flow. The nonlinear evolution of the free surface is then studied numerically. For falling films, increasing the slip length reduces the critical thickness required for plug formation to occur. Families of travelling wave solutions are found via continuation and are used to derive a simple formula for the dependence of this critical thickness on the slip length; this formula is shown to hold for small slip length. For air-driven films, the topology of streamlines in the film can be altered by slip at the wall; if the slip length is large enough, it can prevent regions of recirculation from forming at the wave crest.

Real-time tsunami forecasting system with nonlinear effects using Green’s functions: application to near-shore tsunami behavior in complex bay topography

ABSTRACT In this study, a real-time tsunami estimation method was developed using the linear superposition of Green’s functions, which account for nonlinear tsunami effects around coastal areas. To this end, first, the nonlinear effects associated with the inundation and bottom friction and advection terms in a nonlinear long-wave model were investigated, focusing on large tsunamis around Urado Bay in Kochi Prefecture, Japan, which is at risk of future damage from tsunamis produced by the Nankai Trough earthquake. The results indicate that the effects of advection play a more significant role than those of the inundation and bottom friction in terms of accurately estimating near-shore tsunami propagation. Then, based on these results, Green’s functions were modified to account for the linearly approximated effects of the advection terms. By applying the developed method to tsunamis around Urado Bay, it was demonstrated that accurate estimations of the water surface level and flow velocity changes could be quickly obtained at any location around the bay. The developed method can be applied to real-time tsunami inundation forecasting and off-shore evacuation simulations of vessels to support optimal evacuation decisions.

Resolving indoor shortwave and longwave human body irradiance variations for mean radiant temperature and local thermal comfort

A spatially and directionally resolved longwave and shortwave radiant heat transfer model is presented via a series of experiments in a thermal lab to input surface temperatures and geometries, as well as skin temperature readings from a human subject, in order to test mean radiant temperature (MRT) and thermal comfort results for the person. Combining novel scanning and thermography methods together with ray-tracing simulation, high-resolution thermal models are derived fully characterizing the longwave and shortwave radiant heat fluxes in space and resolving the impact of these variations on MRT. The study demonstrates the significant amount of spatial variation of both shortwave and longwave radiant heat transfer on MRT through the room and also across body segments: the experimental results show variations of up to 14.5 °C across the room, leading to PMV comfort variations from −0.27 to 2.45, clearly demonstrating the importance of mapping the entire radiant field rather than assuming one MRT value for a thermal zone. Furthermore, local radiant temperature, newly defined Body Segment Plane Radiant Temperature (BSPRT), variations across the body of more than 30 °C are found. Finally, a detailed human thermo-physiology model was used to evaluate the possible variation in thermal sensation between the different body segments due to the large differences in local MRT.

A simplified approach for efficiently simulating submarine slump generated tsunamis

A simplified approach was proposed to efficiently simulate the tsunamis generated by a submarine slump. The landslide tsunami generation process was simulated using a long-wave model, simplifying the wave generation problem as a slump traveling down a plane slope with a prescribed trajectory. As a result, the landslide tsunami generation process was parameterized by 11 input parameters. The wave profile at the end of the wave generation process can then be specified as the initial conditions in any numerical tsunami propagation model to study the subsequent tsunami propagation. To demonstrate the capability of this new landslide tsunami generation approach, we used it in combination with an existing Boussinesq wave solver to simulate the 1998 Papua New Guinea landslide tsunami. The results based on the newly calculated physics-based wave profile compare reasonably well with field measurements and the results based on an existing tuning-based wave profile. Sensitivity tests were performed to examine the sensitivity of the runup results to each of the 11 input parameters. Six sets of 100-case Monte Carlo experiments were conducted to investigate the propagation of uncertainty from the input parameters to the runup results. Runup uncertainty was found to be approximately 1.5 times the parameter uncertainty, highlighting the uncertain nature of landslide tsunamis.

Sessile drop evaporation in a gap – crossover between diffusion-limited and phase transition-limited regime

We consider the time evolution of a sessile drop of volatile partially wetting liquid on a rigid solid substrate. The drop evaporates under strong confinement, namely, it sits on one of the two parallel plates that form a narrow gap. First, we develop an efficient mesoscopic long-wave description in gradient dynamics form. It couples the diffusive dynamics of the vertically averaged vapour density in the narrow gap to an evolution equation for the profile of the volatile drop. The underlying free energy functional incorporates wetting, interface and bulk energies of the liquid and gas entropy. The model allows us to investigate the transition between diffusion-limited and phase transition-limited evaporation for shallow droplets. Its gradient dynamics character allows for a long-wave as well as a full-curvature formulation. Second, we compare results obtained with the mesoscopic long-wave model to corresponding direct numerical simulations solving the Stokes equation for the drop coupled to the diffusion equation for the vapour as well as to selected experiments. In passing, we discuss the influence of contact line pinning.

Dynamics of co-current gas–liquid film flow through a slippery channel

We consider a thin liquid film in a wide inclined channel being driven by gravity and co-current turbulent gas flow. The bottom plate with which the liquid is in contact with is taken to be slippery, and we impose the classic Navier slip condition at this substrate. Such a setting finds application in technological processes as well as nature (e.g., distillation, absorption, and cooling devices). The gas–liquid problem can be decoupled by making certain reasonable assumptions. Under these assumptions, we solve the gas problem to obtain the tangential and normal stresses acting at the wavy gas–liquid interface for arbitrary waviness. In modeling the liquid layer dynamics, we make use of the stresses computed in the gas problem as inputs to the interface boundary conditions. We develop the long-wave model and the weighted-integral boundary layer (WIBL) model to describe the thin film dynamics. We perform a linear stability of these reduced order models to scrutinize the effect of wall slip, liquid flow rate, and the gas shear on the stability of the flat film solution. It is found that the wall slip promotes the instability of the flat interface. Furthermore, we compute solitary wave solutions of the WIBL model by implementing Keller's pseudo-arc length algorithm on a periodic domain. We observe that the wave speed as well as the wave amplitude are attenuated on incrementing the liquid slip at the substrate. We corroborate these findings with the time-dependent computations of the nonlinear WIBL model.

Hopf instability of a Rayleigh–Taylor unstable thin film heated from the gas side

A thin liquid film located on the underside of a horizontal solid substrate can be stabilized by the Marangoni effect if the liquid is heated at its free surface. Applying long-wave approximation and projecting the velocity and temperature fields onto a basis of low-order polynomials, we derive a dimension-reduced set of three coupled evolution equations where nonlinearities of both the Navier–Stokes and the heat equation are included. We find that in a certain range of fluid parameters and layer depth, the first bifurcation from the motionless state is oscillatory which sets in with a finite but small wave number. The oscillatory branch is determined using a linear stability analysis of the long-wave model, but also by solving the linearized original hydrodynamic equations. Finally, numerical solutions of the reduced nonlinear model equations in three spatial dimensions are presented.

Analysis and simulation of Korteweg-de Vries-Rosenau-regularised long-wave model via Galerkin finite element method

In this paper, a Galerkin finite element method is designed and analyzed to simulate the nonlinear Korteweg-de Vries-Rosenau-regularized long-wave (KdV-RRLW) model. We establish the existence and uniqueness results in H02(Ω) Sobolev space by applying the Banach–Alaoglu theorem. Using appropriate projection, we derive the error estimates of a semidiscrete scheme for the finite element solution of the KdV-RRLW model. Furthermore, a second-order Crank-Nicolson scheme is employed for the temporal discretization and obtain the optimal order of convergence in the maximum norm. Finally, several numerical examples are provided to visualize the nature of the wave phenomena in one and two dimensional spaces and demonstrate the robustness of the developed finite element algorithm.

Particle trajectories in a weakly nonlinear long-wave model on a shear flow

This work centers on the numerical examination of particle trajectories associated with the propagation of long water waves of small but finite amplitude on a background shear flow over a flat bottom. Taking into consideration the assumption that the nonlinear and dispersive effects are small and of the same magnitude, the Boussinesq-type equations for two-dimensional water waves on a background flow with constant vorticity are derived. Restricting attention to waves propagating in a single direction, we find a new version of the Benjamin-Bona-Mahony (BBM) equation which takes into account the effect of vorticity. In order to investigate the particle trajectories of the flow, an approximate velocity field associated with the derivation of the BBM equation over a shear flow is obtained. Several cases of particle paths under surface waves profiles such as solitary waves and periodic traveling waves are examined.

Investigation of Fractional Nonlinear Regularized Long-Wave Models via Novel Techniques

The main goal of the current work is to develop numerical approaches that use the Yang transform, the homotopy perturbation method (HPM), and the Adomian decomposition method to analyze the fractional model of the regularized long-wave equation. The shallow-water waves and ion-acoustic waves in plasma are both explained by the regularized long-wave equation. The first method combines the Yang transform with the homotopy perturbation method and He’s polynomials. In contrast, the second method combines the Yang transform with the Adomian polynomials and the decomposition method. The Caputo sense is applied to the fractional derivatives. The strategy’s effectiveness is shown by providing a variety of fractional and integer-order graphs and tables. To confirm the validity of each result, the technique was substituted into the equation. The described methods can be used to find the solutions to these kinds of equations as infinite series, and when these series are in closed form, they give the precise solution. The results support the claim that this approach is simple, strong, and efficient for obtaining exact solutions for nonlinear fractional differential equations. The method is a strong contender to contribute to the existing literature.

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.