Motion Of Long Waves Research Articles

M-shaped rational, homoclinic breather, kink-cross rational, multi-wave and interactional soliton solutions to the fifth-order Sawada-Kotera equation

In this work, we develop multi-wave, homoclinic breathers, M-shaped rational, 1-kink interactions with M-shaped, periodic-cross rational and kink-cross rational solutions for the fifth-order Sawada-Kotera equation, which represents the motion of long waves under gravity in shallow water, using several ansatz transformations. A three-wave approach is used to identify multi-wave solitons. Additionally, novel forms of exact solutions are constructed using the homoclinic breathers technique. The kink-cross rational and periodic-cross rational solitons are investigated using appropriate transformations. We develop M-shaped solitons and demonstrate their behavior by selecting suitable parameter values. Furthermore, the interactions between kink waves and M-shaped solitons are also studied. The obtained solutions are determined in 3D, 2D, and contour profiles, where the free parameters involved in the solutions are assigned specific values. Their physical relevance is explored to highlight the inner context of tangible incidents in the natural domain. Since these recently discovered solutions contain a few arbitrary constants, they can be used to explain the variation in qualitative characteristics of wave phenomena.

A New Computational Technique for Analytic Treatment of Time‐Fractional Nonlinear Equations Arising in Magneto‐Acoustic Waves

This paper presents the study of time‐fractional nonlinear fifth‐order Korteweg–de Vries equations by utilizing an adequate novel technique, namely, the q‐homotopy analysis transform method. The fifth‐order Korteweg–de Vries equation has got its importance in the study of magneto‐sound propagation in plasma, capillary gravity water waves, and the motion of long waves under the influence of gravity in shallow water. To justify the effectiveness and pertinence of the contemplated technique, we take a look at three examples of the time‐fractional fifth‐order Korteweg–de Vries equations. The q‐homotopy analysis transform method offers us to modulate the range of convergence of the series solution using ℏ, called the auxiliary parameter or convergence control parameter. The study of the fractional behaviour of the considered equations expresses the originality of the presented work. There is a visible variation in the obtained solutions for different fractional orders and which can lead to different consequences for future work. As a future research direction, readers can use the hybrid methodologies merging with our projected scheme to achieve better consequences. Additionally, to validate the precision and reliability of the proposed method, we organized suitable numerical simulations. The obtained findings show that the proposed method is very gratifying and examines the complex nonlinear challenges that arise in science and innovation.

Длинные волны на шельфе юго-западного побережья о. Сахалин

The results of the study of long-wave motions with periods of more than 20 hours on the shelf of the southwest coast of Sakhalin Island using the time series of sea level fluctuations obtained in full-scale experiments with a discreteness of 1 s and a duration of 4 to 6 months are considered. Spectral analysis of the time series of sea level fluctuations for the period range from 8 to 200 hours revealed the presence of long-wave processes with periods from 26.1 to 46.7 hours, which significantly exceed the inertial period of 16.48 hours. Numerical modeling of shelf waves for exponentially convex profiles of the seabed, carried out using the dispersion relation of V.T. Buchwald and J. K. Adams for waves of the continental shelf, showed that the detected wave processes with periods from 31.2 h to 46.7 h are shelf waves. Their amplitudes increase during storms. The possibility of energy transferring from atmospheric disturbances to shelf waves, which contribute to the formation of sea level, is shown, that confirms the earlier made assumption. Using the calculation of the phase difference of the observed shelf waves at a distance of 12.4 km between Nevelsk and Gornazovodsk and determined by means of the theoretical model, it is shown that the second mode of the shelf wave with a frequency of 0.152 cycle/h is close to the theoretical one. It is shown for the wave with a period of 26.1 h recorded in Ilyinsky and Gornozavodsk for the distance between the points of 173.6 km, that this wave cannot be a shelf wave, but it is a Kelvin wave. This is confirmed by the calculated dispersion diagram, according to which the wavelength of about 689 km corresponds well to the phase difference for the distance Ilyinsky – Gornozavodsk. It is found that shelf waves, one of the generation mechanisms of which is the wind tension along the coast, have the different amplitudes in summer and winter, that is due to the seasonal direction of the offshore wind. In summer, the directions of shelf wave propagation and wind are opposite, which weakens the shelf waves.

Analytic Computational Method for Solving Fractional Nonlinear Equations in Magneto-Acoustic Waves

In this article, we employ a useful and intriguing method known as the ARA-homotopy transform approach to explore the fifth-order Korteweg-de Vries equations that are nonlinear and time-fractional. The study of capillary gravity water waves, magneto-sound propagation in plasma, and the motion of long waves under the effect of gravity in shallow water have all been influenced by Korteweg-de Vries equations. We discuss three instances of the fifth-order time-fractional Korteweg-de Vries equations to demonstrate the efficacy and applicability of the proposed method. Utilizing, also known as the auxiliary parameter or convergence control parameter, the ARA-homotopy transform technique which is a combination between ARA transform and the homotopy analysis method, allows us to modify the convergence range of the series solution. The obtained results show that the proposed method is very gratifying and examines the complex nonlinear challenges that arise in science and innovation.

One-soliton shaping and inelastic collision between double solitons in the fifth-order variable-coefficient Sawada–Kotera equation

The main concern of the present article is to study the fifth-order variable-coefficient Sawada–Kotera (VcSK) equation which describes the motion of long waves in shallow water under the gravity. A single- and double-soliton rational solutions for this model are formally retrieved through the generalized unified method. For a single-soliton wave, the velocity, the amplitude and the shape of the wave cannot be affected by variable coefficients. There is an inelastic collision (the collision that makes change in amplitude of the soliton waves and shifts in their trajectories) between the double-soliton waves due to the time-varying field in a graded-index waveguide. It hoped that the established solutions can be used to enrich the dynamic behaviors of the VcSK equation.

Control of a Boussinesq system of KdV–KdV type on a bounded interval

We consider a Boussinesq system of KdV–KdV type introduced by J.L. Bona, M. Chen and J.-C. Saut as a model for the motion of small amplitude long waves on the surface of an ideal fluid. This system of two equations can describe the propagation of waves in both directions, while the single KdV equation is limited to unidirectional waves. We are concerned here with the exact controllability of the Boussinesq system by using some boundary controls. By reducing the controllability problem to a spectral problem which is solved by using the Paley–Wiener method introduced by the third author for KdV, we determine explicitly all the critical lengths for which the exact controllability fails for the linearized system, and give a complete picture of the controllability results with one or two boundary controls of Dirichlet or Neumann type. The extension of the exact controllability to the full Boussinesq system is derived in the energy space in the case of a control of Neumann type. It is obtained by incorporating a boundary feedback in the control in order to ensure a global Kato smoothing effect.

Asymptotic behavior of Boussinesq system of KdV–KdV type

This work deals with the local rapid exponential stabilization for a Boussinesq system of KdV–KdV type introduced by J. Bona, M. Chen and J.-C. Saut. This is a model for the motion of small amplitude long waves on the surface of an ideal fluid. Here, we will consider the Boussinesq system of KdV–KdV type posed on a finite domain, with homogeneous Dirichlet–Neumann boundary controls acting at the right end point of the interval. Our goal is to build suitable integral transformations to get a feedback control law that leads to the stabilization of the system. More precisely, we will prove that the solution of the closed-loop system decays exponentially to zero in the L2(0,L)-norm and the decay rate can be tuned to be as large as desired if the initial data is small enough.

Lie point symmetries, conservation laws, and analytical solutions of a generalized time-fractional Sawada–Kotera equation

ABSTRACTUnder investigation in this work is the invariance properties of the time-fractional generalized Sawada–Kotera equation, which can describe motion of long waves in shallow water under gravity and in a one-dimensional nonlinear lattice. With the help of the Lie symmetry analysis method of fractional differential equations, we strictly derive the vector fields and symmetry reductions of the equation. Furthermore, based on the power series theory, an effective method is presented to succinctly construct its analytical solutions. Finally, using the new conservation theorem, the conservation laws of the equation are well constructed with a detailed derivation.

Calculation of the sea wave run-up on a sloping beach

The motion of long surface waves above a sloping bottom has been calculated within the shallow-water approximation. It is shown that, if the linear solution is known, the nonlinear corrections, up to the eighth order, can be expressed in terms of the derivatives of the functions determining the linear solution. An analytical solution for the free-surface elevation in the near-shore area is constructed.

Energy conservative solutions to the system of full variational sine-Gordon equations in a unit sphere

We establish the global existence of a conservative weak solution to the Cauchy problem for a complete system of variational sine-Gordon equations, which models the motion of long waves on a neutral dipole chain in the continuum limit in a unit sphere. Although singularities may develop in finite time, the energy of the solution is conserved across singular times. We also obtain the continuous dependence of solutions on the given initial data.

Weak solutions to a nonlinear variational sine–Gordon equation

This paper considers a nonlinear variational sine–Gordon equation utt−c(u)[c(u)ux]x+λ22sin(2u)=0 which describes the motion of long waves on a neutral dipole chain in the continuum limit and a few other physical phenomena. We establish the global existence of dissipative weak solutions to its Cauchy problem for initial data in the space H1×L2 by using the Young measure theory in the setting of Lp spaces.

Long-wave dispersion phenomena in a layer subject to elastically restrained boundary conditions

A generalisation of the usual free and fixed-face boundary conditions for an elastic layer is considered. Specifically, this is in the context of a linear isotropic elastic layer and involves an investigation into long-wave motion. Elastically restrained boundary conditions are introduced, with both a numerical investigation and a multi-parameter asymptotic analysis employed to elucidate dispersion phenomena in a layer subject to such boundary conditions. The characteristics of mode transformations associated with the change from traction-free to fixed-face boundary conditions are described, focusing particularly on the fundamental modes which are known to not occur in the case of fixed faces. All possible asymptotic regimes are grouped into six classes, with the corresponding expansions of frequency in respect of wave number derived.

Interaction of two sea surface waves over a sloping bottom

A nonlinear model is constructed for the motion of long surface waves over a sloping bottom in the shallow-water approximation. It is shown that if a linear solution is known, the quadratic and cubic nonlinear corrections can be expressed in terms of the derivatives of the functions determining the linear solution. The developed method is applied to the interaction of two periodic waves in the near-shore area.

Conservative solutions to a nonlinear variational sine-Gordon equation

This paper is concerned with a nonlinear variational sine-Gordon equation u t t − c ( u ) [ c ( u ) u x ] x + λ 2 2 s i n ( 2 u ) = 0 which describes the motion of long waves on a neutral dipole chain in the continuum limit and a few other physical phenomena. We establish the global existence of an energy-conservative weak solution to its Cauchy problem for initial data of finite energy. To deal with the possible blowup of solutions, we introduce a new set of variables depending on the energy, whereby all singularities are resolved.

Alongshore fluid motions in the swash zone of a sandy and gravel beach

Field experiments were conducted on a low-gradient, high-energy sandy beach (Truc Vert, France) and a steep, low-energy gravel beach (Slapton, UK) to examine alongshore-directed currents within the swash zone. At Truc Vert, data were collected over 33 tidal cycles with offshore significant wave heights of 1–4 m and periods of 5–12 s. At Slapton data were collected during 12 tides with wave heights of 0.3–1 m and periods of 4–9 s. The swash motion was predominantly at infragravity frequencies at Truc Vert and incident frequencies at Slapton. During selected tides, strong alongshore currents of the same order of magnitude as the cross-shore flows were a consistent feature of both swash zones; however, while the cross-shore flows were saw-toothed with a distinct flow reversal, the alongshore flows were unidirectional and continuous. On the sand beach, alongshore momentum was advected into the swash from the surf zone by standing long-wave motion resulting in the strongest flows in the lower swash. In contrast on the gravel beach, highly oblique uprush directly forced the longshore-directed motions and resulted in strong flow velocities in the upper swash. On the sandy beach, the cross- and longshore boundary layers were consistently different with strong vertical shear during the uprush for the cross-shore and during the backwash for the alongshore flows. A logarithmic profile better fitted the longshore flows during the backwash, compared to during the uprush for the cross-shore flows, and enabled the drag coefficient to be computed. The alongshore drag coefficient was C d, v = 10 − 3 and of the same order as for the cross-shore, but was generally larger during the backwash compared to the uprush. Sediment transport efficiency during the backwash is potentially enhanced by longshore-directed swash flows. Significant bed shear stress is maintained throughout that cross-shore flow reversal enabling down-slope transport earlier in the swash cycle.

Coupling of Hydrodynamic and Wave Models: Case Study for Hurricane Floyd (1999) Hindcast

This paper demonstrates a practical application of coupling a hydrodynamic model with a wave model for the calculation of storm tide elevations in the St. Johns River (Northeastern Florida). Hurricane Floyd (1999) is chosen as the storm of interest due to its track which paralleled the northeast coast of Florida without making a direct landfall on the St. Johns River. The advanced circulation (ADCIRC) numerical code is applied as the hydrodynamic model for the computation of two-dimensional circulation resulting from astronomic tides and meteorologically induced storm surge. The simulating waves nearshore (SWAN) numerical code is applied as the wave model for the computation of wind-induced waves. Two model implementations are considered in order to investigate the effect of short wave contributions on the overall storm tide water level: (1) a one-way coupling procedure that transfers gradient of wave radiation stresses from SWAN to ADCIRC; and (2) a two-way coupling procedure that builds on the former to include feedback of water levels and currents from ADCIRC to SWAN. Simulated storm tide elevations are compared to historical National Ocean Service data to result in two major conclusions: (1) wind-induced waves play a significant role in the storm tide (contributing 10–15% of the peak water level), namely with respect to the transfer of momentum from the dissipation of short waves to the long-wave motion of the storm surge; and (2) a local-scale hydrodynamic model requires the application of a hydrograph boundary condition in order to account for the remote effects of the storm surge response.

Lagrangian and Hamiltonian two-scale reduction

Studying spatially extended Hamiltonian systems with coherent microstructure, it is an important and challenging problem to identify reduced Hamiltonian models that describe the effective dynamics on large spatial and temporal scales. Such models require macroscopically varying, deterministic initial data which can possess a well-organized microstructure. This paper concerns the question how reasonable macroscopic Lagrangian and Hamiltonian structures can be derived from the microscopic system. In the first part we develop a general approach by considering noncanonical Hamiltonian structures on the tangent bundle. This approach can be applied to all Hamiltonian lattices [or Hamiltonian partial differential equations (PDEs)] and involves three building blocks: (i) the embedding of the microscopic system, (ii) an invertible two-scale transformation that encodes the underlying scaling of space and time, and (iii) an elementary model reduction that is based on a principle of consistent expansions. The result is always a reduced Hamiltonian PDE that governs the macroscopic dynamics, provided that the initial data are chosen consistently with the underlying two-scale ansatz. In the second part we exemplify the reduction approach and derive various reduced PDE models for the atomic chain. The reduced equations are either related to long-wave motion or describe macroscopic modulations of oscillatory microstructures.

The influence of an initial simple shear deformation on long-wave motion in an elastic layer

Long wave dispersion in an incompressible elastic layer subject to an initial static simple shear deformation is investigated. Long wave approximations of the dispersion relation associated with zero incremental traction on the faces are derived for both low and high-frequency motion. Comparison of approximate and numerical solutions is shown to provide excellent agreement over a surprisingly large wave number range. Within both the low and high-frequency regimes, the approximations are employed to establish the relative asymptotic orders of the displacement components and hydrostatic pressure. In the high-frequency case, the in-plane component of displacement is shown to be asymptotically larger than the normal component; motion is, therefore, essentially that of thickness shear resonance. The influence of this specific form of initial deformation is, therefore, seemingly minor in respect of longwave high-frequency motion. However, in the long-wave low-frequency case, considerable differences are noted in comparison with both the classical and previously published prestressed cases. Specifically, both the normal and in-plane displacement components are of the same asymptotic order, indicating the absence of any natural analogue of either classical bending or extension.

Hamiltonian long wave expansions for internal waves over a periodically varying bottom

We derive a Hamiltonian formulation for two-dimensional nonlinear long waves between two bodies of immiscible fluid with a periodic bottom. From the formulation and using the Hamiltonian perturbation theory, we obtain effective Boussinesq equations that describe the motion of bidirectional long waves and unidirectional equations that are similar to the KdV equation for the case in which the bottom possesses short length scale. The computations for these results are performed in the framework of an asymptotic analysis of multiple scale operators.

Fluid–structure interaction between a two-dimensional mat-type VLFS and solitary waves by the Green–Naghdi theory

The two-dimensional, nonlinear hydroelasticity of a mat-type very large floating structure (VLFS) is studied within the scope of linear beam theory for the structure and the nonlinear, Level I Green–Naghdi (GN) theory for the fluid. The beam equation and the GN equations are coupled through the kinematic and dynamic boundary conditions to obtain a new set of modified GN equations. These equations represent long-wave motion beneath an elastic plate. A set of jump conditions that are necessary for the continuity (or the matching) of the solutions in the open water region and that under the structure is derived through the use of the postulated conservation laws of mass, momentum, and mechanical energy. The resulting governing equations, subjected to the boundary and jump conditions, are solved by the finite-difference method in the time domain. The present model is applicable, for example, to the study of the hydroelastic response of a mat-type VLFS under the action of a solitary wave, or a frontal tsunami wave. Good agreement is observed between the model results and other published theoretical and numerical predictions, as well as experimental data. The results show that consideration of nonlinearity is important for accurate predictions of the bending moment of the floating elastic plate. It is found that the rigidity of the structure greatly affects the bending moment and displacement of the structure in this nonlinear theory.