Forced Korteweg-de Vries Equation Research Articles

Soliton interaction with external forcing within the Korteweg-de Vries equation.

We revise the solutions of the forced Korteweg-de Vries equation describing a resonant interaction of a solitary wave with external pulse-type perturbations. In contrast to previous work where only the limiting cases of a very narrow forcing in comparison with the initial soliton or a very narrow soliton in comparison with the width of external perturbation were studied, we consider here an arbitrary relationship between the widths of soliton and external perturbation of a relatively small amplitude. In many particular cases, exact solutions of the forced Korteweg-de Vries equation can be obtained for the specific forcings of arbitrary amplitude. We use the earlier developed asymptotic method to derive an approximate set of equations up to the second-order on a small parameter characterising the amplitude of external force. The analysis of exact solutions of the derived equations is presented and illustrated graphically. It is shown that the theoretical outcomes obtained by the asymptotic method are in a good agreement with the results of direct numerical modeling within the framework of forced Korteweg-de Vries equation.

Optimal Shape of an Underwater Moving Bottom Generating Surface Waves Ruled by a Forced Korteweg-de Vries Equation

It is well known since Wu and Wu (in: Proceedings of the 14th symposium on naval hydrodynamics, National Academy Press, Washington, pp 103–125, 1982) that a forcing disturbance moving steadily with a transcritical velocity in shallow water can generate, continuously and periodically, a succession of solitary waves propagating ahead of the disturbance in procession. One possible new application of this phenomenon could very well be surfing competitions, where in a controlled environment, such as a pool, waves can be generated with the use of a translating bottom. In this paper, we use the forced Korteweg–de Vries equation to investigate the shape of the moving body capable of generating the highest first upstream-progressing solitary wave. To do so, we study the following optimization problem: maximizing the total energy of the system over the set of non-negative square-integrable bottoms, with uniformly bounded norms and compact supports. We establish analytically the existence of a maximizer saturating the norm constraint, derive the gradient of the functional, and then implement numerically an optimization algorithm yielding the desired optimal shape.

Bimodal pair f-KdV dynamics in star-forming clouds

A theoretical formalism for investigating the bimodal conjugational mode dynamics of hybrid source, dictated by a unique pair of forced Korteweg-de Vries (f-KdV) equations in a complex turbo-magnetized star-forming cloud, is reported. It uses a standard multi-scale analysis executed over the cloud-governing equations in a closure form to derive the conjugated pair f-KdV system. We numerically see the structural features of two distinctive classes of eigenmode patterns stemming from the conjoint gravito-electrostatic interplay. The electrostatic compressive monotonic aperiodic shock-like patterns and gravitational compressive non-monotonic oscillatory shock-like structures are excitable. It is specifically revealed that the constitutive grain-charge (grain-mass) acts as electrostatic stabilizer (gravitational destabilizer) against the global cloud collapse dynamics. The basic features of the nonlinear coherent structures are confirmed in systematic phase-plane landscapes, indicating electrostatic irregular non-homoclinic open trajectories and gravitational atypical non-chaotic homoclinic fixed-point attractors. The relevance in the real astro-cosmic scenarios of the early phases of structure formation via wave-driven fluid-accretive transport processes is summarily emphasized.

Internal solitary wave generation by tidal flow over topography

Oceanic internal solitary waves are typically generated by barotropic tidal flow over localised topography. Wave generation can be characterised by the Froude number $F=U/c_{0}$, where $U$ is the tidal flow amplitude and $c_{0}$ is the intrinsic linear long wave phase speed, that is the speed in the absence of the tidal current. For steady tidal flow in the resonant regime, $\unicode[STIX]{x1D6E5}_{m}<F-1<\unicode[STIX]{x1D6E5}_{M}$, a theory based on the forced Korteweg–de Vries equation shows that upstream and downstream propagating undular bores are produced. The bandwidth limits $\unicode[STIX]{x1D6E5}_{m,M}$ depend on the height (or depth) of the topographic forcing term, which can be either positive or negative depending on whether the topography is equivalent to a hole or a sill. Here the wave generation process is studied numerically using a forced Korteweg–de Vries equation model with time-dependent Froude number, $F(t)$, representative of realistic tidal flow. The response depends on $\unicode[STIX]{x1D6E5}_{max}=F_{max}-1$, where $F_{max}$ is the maximum of $F(t)$ over half of a tidal cycle. When $\unicode[STIX]{x1D6E5}_{max}<\unicode[STIX]{x1D6E5}_{m}$ the flow is always subcritical and internal solitary waves appear after release of the downstream disturbance. When $\unicode[STIX]{x1D6E5}_{m}<\unicode[STIX]{x1D6E5}_{max}<\unicode[STIX]{x1D6E5}_{M}$ the flow reaches criticality at its peak, producing upstream and downstream undular bores that are released as the tide slackens. When $\unicode[STIX]{x1D6E5}_{max}>\unicode[STIX]{x1D6E5}_{M}$ the tidal flow goes through the resonant regime twice, producing undular bores with each passage. The numerical simulations are for both symmetrical topography, and for asymmetric topography representative of Stellwagen Bank and Knight Inlet.

Analytical Solitary Wave Solution of the Dust Ion Acoustic Waves for the Damped Forced Korteweg–de Vries Equation in Superthermal Plasmas

Abstract Analytical solitary wave solution of the dust ion acoustic (DIA) waves was studied in the framework of the damped forced Korteweg–de Vries (DFKdV) equation in superthermal collisional dusty plasmas. The reductive perturbation technique was applied to derive the DKdV equation. It is observed that both the rarefactive and compressive solitary wave solutions are possible for this plasma model. The effects of κ and the strength (f 0) and frequency (ω) of the external periodic force were studied on the analytical solitary wave solution of the DIA waves. It is observed that the parameters κ, f 0 and ω have significant effects on the structure of the damped forced DIA solitary waves. The results of this study may have relevance in laboratory plasmas as well as in space plasmas.

A study of wave trapping between two obstacles in the forced Korteweg–de Vries equation

In the forced Korteweg–de Vries equation, a trail of waves can be temporarily trapped between two bumps. Wave propagations vary depending on the solution types and the form of the bumps. In this paper, we study the propagation tendencies of these temporarily trapped waves and especially consider the solitary and flat top solutions over the bumps of different distances and heights. A concept of energy related to wave speeds is defined in the region between the bumps, and it turns out that the trapped waves have to overcome a certain threshold of energy in order to escape out of the region. The numerical investigations are focused on the influence of the distance and the height of the bumps on the energy barrier.

Spacial inhomogeneity and nonlinear tunneling for the forced KdV equation

A variable-coefficient forced Korteweg–de Vries equation with spacial inhomogeneity is investigated in this paper. Under constraints, this equation is transformed into its bilinear form, and multi-soliton solutions are derived. Effects of spacial inhomogeneity for soliton velocity, width and background are discussed. Nonlinear tunneling for this equation is presented, where the soliton amplitude can be amplified or compressed. Our results might be useful for the relevant problems in fluids and plasmas.

Higher-order modulation theory for resonant flow over topography

The flow of a fluid over isolated topography in the long wavelength, weakly nonlinear limit is considered. The upstream flow velocity is assumed to be close to a linear long wave velocity of the unforced flow so that the flow is near resonant. Higher order nonlinear, dispersive and nonlinear-dispersive terms beyond the Korteweg-de Vries approximation are included so that the flow is governed by a forced extended Korteweg-de Vries equation. Modulation theory solutions for the undular bores generated upstream and downstream of the forcing are found and used to study the influence of the higher-order terms on the resonant flow, which increases for steeper waves. These modulation theory solutions are compared with numerical solutions of the forced extended Korteweg-de Vries equation for the case of surface water waves. Good comparison is obtained between theoretical and numerical solutions, for properties such as the upstream and downstream solitary wave amplitudes and the widths of the bores.

New Aspects of Exponential Asymptotics in Multiple‐Scale Nonlinear Wave Problems

It is known that standard multiple‐scale perturbation techniques fail to pinpoint the soliton solution branches that bifurcate at edges of bandgaps in periodic media, owing to the appearance of exponentially small growing wave tails when the soliton's envelope is not properly positioned. When the bifurcation is from a single wave mode of a band edge, this difficulty has been handled in recent work by computing these tails via an exponential asymptotics technique in the wave number domain. However, the same approach is not directly applicable to the bifurcation of solitons near the opening of a bandgap, where wave modes from two nearby band edges interact with each other. Here, we discuss two nontrivial extensions of the exponential asymptotics technique that enable resolving this issue. For simplicity, the analysis focuses on two model problems, namely, a steady‐state forced Korteweg–de Vries equation and a steady‐state forced nonlinear Schrödinger equation, with the precise form of forcing and balance between nonlinear and dispersive terms chosen so as to mimic the situation encountered in the bifurcation of solitons near a bandgap opening. Our analysis exhibits a number of new features that are significantly different from previous exponential asymptotics procedures, such as the treatments when the nonlinearity dominates dispersion and when the decay rates of the Fourier‐transformed solution are asymmetric. In addition, the analysis reveals new, and in some cases rather unexpected, functional forms for exponentially small wave tails, which are also confirmed by numerical results.

Radiation of short waves from the resonantly excited capillary–gravity waves

Capillary–gravity waves resonantly excited by an obstacle (Froude number: $Fr=1$) are investigated by the numerical solution of the Euler equations. The radiation of short waves from the long nonlinear waves is observed when the capillary effects are weak (Bond number: $Bo<1/3$). The upstream-advancing solitary wave radiates a short linear wave whose phase velocity is equal to the solitary waves and group velocity is faster than the solitary wave (soliton radiation). Therefore, the short wave is observed upstream of the foremost solitary wave. The downstream cnoidal wave also radiates a short wave which propagates upstream in the depression region between the obstacle and the cnoidal wave. The short wave interacts with the long wave above the obstacle, and generates a second short wave which propagates downstream. These generation processes will be repeated, and the number of wavenumber components in the depression region increases with time to generate a complicated wave pattern. The upstream soliton radiation can be predicted qualitatively by the fifth-order forced Korteweg–de Vries equation, but the equation overestimates the wavelength since it is based on a long-wave approximation. At a large Bond number of $Bo=2/3$, the wave pattern has the rotation symmetry against the pattern at $Bo=0$, and the depression solitary waves propagate downstream.

Transcritical flow over two obstacles: forced Korteweg–de Vries framework

We consider free-surface flow over two localised obstacles using the framework of the forced Korteweg–de Vries equation in a suite of numerical simulations. Our main concern is with the transcritical regime when the oncoming flow has a Froude number close to unity. The flow behaviour can be characterised by the Froude number and the maximum heights of the obstacles. In the transcritical regime at early times, undular bores are produced upstream and downstream of each obstacle. Our main aim is to describe the interaction of these undular bores between the obstacles, and to find the outcome at very large times. We find that the flow development can be defined in three stages. The first stage is described by the well-known development of undular bores upstream and downstream of each obstacle. The second stage is the interaction between the undular bore moving downstream from the first obstacle and the undular bore moving upstream from the second obstacle. The third stage is the very large time evolution of this interaction, when one of the obstacles controls criticality. For equal obstacle heights, our analytical and numerical results indicate that either one of the obstacles can control flow criticality, that being the first obstacle when the flow is slightly subcritical and the second obstacle otherwise. For unequal obstacle heights the larger obstacle controls criticality. The results obtained here complement a recent numerical study using the fully nonlinear, but non-dispersive, shallow water equations.

Wakes and precursor soliton excitations by a moving charged object in a plasma

We study the evolution of nonlinear ion acoustic wave excitations due to a moving charged source in a plasma. Our numerical investigations of the full set of cold fluid equations go beyond the usual weak nonlinearity approximation and show the existence of a rich variety of solutions including wakes, precursor solitons, and “pinned” solitons that travel with the source velocity. These solutions represent a large amplitude generalization of solutions obtained in the past for the forced Korteweg deVries equation and can find useful applications in a variety of situations in the laboratory and in space, wherever there is a large relative velocity between the plasma and a charged object.

Steady free-surface flow over spatially periodic topography

Two-dimensional free-surface flow over a spatially periodic channel bed topography is examined using a steady periodically forced Korteweg–de Vries equation. The existence of new forced solitary-type waves with periodic tails is demonstrated using recently developed non-autonomous dynamical-systems theory. Bound states with two or more co-existing solitary waves are also identified. The solution space for varying amplitude of forcing is explored using a numerical method. A rich bifurcation structure is uncovered and shown to be consistent with an asymptotic theory based on small forcing amplitude.

Trapped supercritical waves for the forced KdV equation with two bumps

The forced Korteweg–de Vries equation (fKdV) is considered to study free surface flows in a two-dimensional channel over an obstacle. The forcing is due to a bottom topography, where we consider one bump and two bumps. In the study of the wave motion of the free surface, there is an important parameter value, the Froude number (F), the ratio of the upstream velocity to the critical speed of shallow water wave. Our focus is on the supercritical case (F>1), where solitary wave solutions can be observed. Interesting surface wave phenomena can be observed when a bottom topography is rather complicated. First, we revisit various stationary wave solutions of the forced KdV equation in the presence of two bumps. There are multiple trapped supercritical wave solutions between two bumps. Their numerical stability is investigated under different bump scenarios. Our numerical results show that the interplay between trapped solitary waves and two bumps plays a key role to determine the time evolution of those wave solutions. Interestingly, there are multiple trapped supercritical wave solutions, which are trapped between two bumps up to a certain time.

Phase-modulated solitary waves controlled by a boundary condition at the bottom.

A forced Korteweg-de Vries (KdV) equation is derived to describe weakly nonlinear, shallow-water surface wave propagation over nontrivial bottom boundary condition. We show that different functional forms of bottom boundary conditions self-consistently produce different forced KdV equations as the evolution equations for the free surface. Solitary wave solutions have been analytically obtained where phase gets modulated controlled by bottom boundary condition, whereas amplitude remains constant.

A Locally Conservative Eulerian–Lagrangian Finite Difference Method for the forced KdV equation

We propose a Locally Conservative Eulerian–Lagrangian Finite Difference Method (fdLCELM) for approximating the solution of the forced Korteweg–de Vries equation. The new numerical method employs an operator splitting scheme that solves the nonlinear transport equation and the linear dispersive equation sequentially. In order to conserve mass in the transport fractional step, we trace back each grid cell in time along the integral curve. The dispersive fractional step will be solved using a cell-centered finite difference method.Numerical examples are provided to confirm and illustrate the accuracy and mass conservation property of the new method. We also study the numerical stability and time evolution of various stationary solitary wave solutions in the presence of one or two bumps.

A simple and robust boundary treatment for the forced Korteweg–de Vries equation

In this paper, we propose a simple and robust numerical method for the forced Korteweg–de Vries (fKdV) equation which models free surface waves of an incompressible and inviscid fluid flow over a bump. The fKdV equation is defined in an infinite domain. However, to solve the equation numerically we must truncate the infinite domain to a bounded domain by introducing an artificial boundary and imposing boundary conditions there. Due to unsuitable artificial boundary conditions, most wave propagation problems have numerical difficulties (e.g., the truncated computational domain must be large enough or the numerical simulation must be terminated before the wave approaches the artificial boundary for the quality of the numerical solution). To solve this boundary problem, we develop an absorbing non-reflecting boundary treatment which uses outward wave velocity. The basic idea of the proposing algorithm is that we first calculate an outward wave velocity from the solutions at the previous and present time steps and then we obtain a solution at the next time step on the artificial boundary by moving the solution at the present time step with the velocity. And then we update solutions at the next time step inside the domain using the calculated solution on the artificial boundary. Numerical experiments with various initial conditions for the KdV and fKdV equations are presented to illustrate the accuracy and efficiency of our method.

Transcritical flow of a stratified fluid over topography: analysis of the forced Gardner equation

AbstractTranscritical flow of a stratified fluid past a broad localised topographic obstacle is studied analytically in the framework of the forced extended Korteweg–de Vries, or Gardner, equation. We consider both possible signs for the cubic nonlinear term in the Gardner equation corresponding to different fluid density stratification profiles. We identify the range of the input parameters: the oncoming flow speed (the Froude number) and the topographic amplitude, for which the obstacle supports a stationary localised hydraulic transition from the subcritical flow upstream to the supercritical flow downstream. Such a localised transcritical flow is resolved back into the equilibrium flow state away from the obstacle with the aid of unsteady coherent nonlinear wave structures propagating upstream and downstream. Along with the regular, cnoidal undular bores occurring in the analogous problem for the single-layer flow modelled by the forced Korteweg–de Vries equation, the transcritical internal wave flows support a diverse family of upstream and downstream wave structures, including kinks, rarefaction waves, classical undular bores, reversed and trigonometric undular bores, which we describe using the recent development of the nonlinear modulation theory for the (unforced) Gardner equation. The predictions of the developed analytic construction are confirmed by direct numerical simulations of the forced Gardner equation for a broad range of input parameters.

On the second-order approximate symmetry classification and optimal systems of subalgebras for a forced Korteweg–de Vries equation

We investigate a forced Korteweg–de Vries (fKdV) equation, u,t+cu,x+αuu,x+βu,xxx=βF(t), which arises in the modelling of tsunami generation by submarine landslides. Approximate symmetries are found for the fKdV equation using the method as proposed by Fushchich and Shtelen [6]. Symmetries are found to second order in the perturbation parameter using the MAPLE symmetry package ASP[11], an add-on to the symmetry package DESOLVII[18]. ASP also allows particular forms of the arbitrary function F(t) to be found which extend the symmetry algebra and hence a full approximate symmetry classification of the fKdV equation is obtained. Optimal systems of one-dimensional subalgebras are also determined. Corresponding approximate invariant solutions to the fKdV equation are then constructed for particular F(t) using DESOLVII routines.

Lie Group Analysis of a Forced KdV Equation

The Korteweg-de Vries (KdV) equation considered in this work contains a forcing term and is referred to as forced KdV equation in the sequel. This equation has been investigated recently as a mathematical model for waves on shallow water surfaces under the influence of external forcing. We employ the Lie group analysis approach to specify the time-dependent forcing term.