Forced Korteweg-de Vries Equation Research Articles

A gradient reproducing kernel based stabilized collocation method for the 5th order Korteweg–de Vries equations

This paper presents a gradient reproducing kernel based stabilized collocation method (GRK-SCM) to solve the generalized nonlinear 5th order Korteweg–de Vries (KdV) equations. By introducing gradient reproducing kernel (GRK) approximations, one can circumvent the intricacy of high-order derivatives in RK approximations, while still satisfying high-order consistency requirements. This leads to the high accuracy and efficiency. GRKs are very good approximation candidates for addressing high order partial differential equations that require higher order derivatives formulated as strong form. The stabilized collocation method (SCM) effectively meets high-order integration constraints which can achieve accurate integration in the subdomains. This ensures both high precision and optimal convergence. Von Neumann analysis is utilized to establish the stability criteria for GRK-SCM when combined with forward difference temporal discretization. Numerical solutions for Sawada–Kotera (SK) equation and Kaup-Kupershmidt (KK) equation are studied where the solitary wave migration and collision, and periodic waves are represented. A fifth order forced KdV equation is also considered. The presented method's high accuracy and convergence are demonstrated through comparative studies with analytical solutions, and investigations of invariants and corresponding errors determine the good conservation properties of this algorithm.

A compact finite difference scheme with absorbing boundary condition for forced KdV equation

Studying the long-time solution behavior of the Korteweg-de Vries (KdV) type equation with a periodic force acting at one end of the long channel is important for simulating the blood flow in artery driven by heart pulses. It is of great interest to develop an accurate numerical method for solving the forced KdV problem. In this article, we present the following methods to obtain an accurate approximation to the solution of KdV problem.•An accurate compact finite difference scheme is proposed for solving the above forced KdV problem with fourth-order accuracy.•An absorbing boundary condition at the right end of the interval is used to avoid the wave reflection.•The stability of scheme is proved by the von Neumann method and then tested by three examples. Results show that the method provides an accurate solution, and the wave propagates without reflection.

Solitary Wave Interactions with an External Periodic Force: The Extended Korteweg-de Vries Framework

In this work we asymptotically and numerically studied the interaction of large amplitude solitary waves with an external periodic force using the forced extended Korteweg-de Vries equation (feKdV). Regarding these interactions, we found three types of regimes depending on the amplitude of the solitary wave and how its speed and the speed of the external force are related. A solitary wave can remain steady when its crest and the crest of the external force are in phase, it can bounce back and forth remaining close to its initial position when its speed and the external force speed are near resonant, or it can move away from its initial position without reversing its direction. Additionally, we verified that the numerical results agreed qualitatively well within the asymptotic approximation theory for external broad forces.

Solitary Wave Solutions for Forced Nonlinear Korteweg-de Vries Equation by Using Approximate Analytical Method

The Multistep Modified Reduced Differential Transform Method (MMRDTM) is proposed in this paper. It is implemented to solve the forced nonlinear Korteweg-de Vries (fnKdV) equations. There are several advantages of the proposed method. Using the suggested approach, a high-speed converging series can be approximated analytically. Furthermore, the number of calculated terms is reduced significantly. The nonlinear term in fnKdV equations is substituted with corresponding Adomian polynomials before applying the multi-step technique. As a result, we provided simpler and more effective ways to solve fnKdV equations. On top of that, the solutions can be approximated more accurately over a longer period of time. To show the MMRDTM's capability and accuracy, we consider several of the fnKdV examples to illustrate the proposed method’s potential in analytical approximation. Then, the features of the solutions are represented in tabular and graphical forms. In conclusion, the proposed method delivers highly accurate and precise solutions for these types of equations.

Nonlinear spectral analysis of ion acoustic solitons arising from a streaming charged object using the numerical inverse scattering transform

Observations of nonlinear coherent plasma phenomena in spacecraft and terrestrial experiments often rely on visual identification of solitary modes because nonlinear coherent modes are broadband in a Fourier transform based analysis. We implement an alternative spectral decomposition known as the inverse scattering transform and demonstrate its ability to successfully isolate nonlinear modes on the homogeneous and forced Korteweg–De Vries equation which models these nonlinear modes. We also demonstrate for the first time that this decomposition is useful when forcing is applied. This is because the stable modes generated by localized forcing are similar to the homogeneous solutions where the inverse scattering transform is a rigorous decomposition. This spectral technique is then applied to simulations of ion acoustic waves generated by a 10 mm spherical debris object interacting with the ionospheric plasma orbiting at 2000 km altitude. The algorithm is found to successfully detect the resultant solitons. This demonstrates the feasibility of using this spectral technique as a real time analysis tool for screening spacecraft data for nonlinear solitary modes.

A computational approach for shallow water forced Korteweg–De Vries equation on critical flow over a hole with three fractional operators

The Korteweg–De Vries (KdV) equation has always provided a venue to study and generalizes diverse physical phenomena. The pivotal aim of the study is to analyze the behaviors of forced KdV equation describing the free surface critical flow over a hole by finding the solution with the help of q-homotopy analysis transform technique (q-HATT). he projected method is elegant amalgamations of q-homotopy analysis scheme and Laplace transform. Three fractional operators are hired in the present study to show their essence in generalizing the models associated with power-law distribution, kernel singular, non-local and non-singular. The fixed-point theorem employed to present the existence and uniqueness for the hired arbitrary-order model and convergence for the solution is derived with Banach space. The projected scheme springs the series solution rapidly towards convergence and it can guarantee the convergence associated with the homotopy parameter. Moreover, for diverse fractional order the physical nature have been captured in plots. The achieved consequences illuminates, the hired solution procedure is reliable and highly methodical in investigating the behaviours of the nonlinear models of both integer and fractional order.

Solitary water wave interactions for the forced Korteweg–de Vries equation

The aim of this work is to study solitary water wave interactions between two topographic obstacles for the forced Korteweg–de Vries equation (fKdV). Focusing on the details of the interactions, we identify regimes in which solitary wave interactions maintain two well separated crests and regimes where the number of local maxima varies according to the laws \(2\rightarrow 1\rightarrow 2\rightarrow 1\rightarrow 2\) or \(2\rightarrow 1\rightarrow 2\). It shows that the geometric Lax-categorization of Korteweg–de Vries (KdV) two-soliton interactions still holds for the fKdV equation.

A numerical method for finding solution of the distributed‐order time‐fractional forced Korteweg–de Vries equation including the Caputo fractional derivative

In this paper, for the first time, the distributed‐order time‐fractional forced Korteweg–de Vries equation is studied. We use a numerical method based on the shifted Legendre operational matrix of distributed‐order fractional derivative with Tau method to find approximate solution of distributed‐order forced Korteweg–de Vries equation. This shifted Legendre operational matrix of distributed‐order fractional derivative with Tau method is used to reduce the solution of the distributed‐order time‐fractional forced Korteweg–de Vries equations to a system of algebraic equations. An error analysis and convergence are obtained. Finally, to display the applicability and validity of the numerical method, some examples are implemented.

Gravity–capillary flows over obstacles for the fifth-order forced Korteweg–de Vries equation

The aim of this work is to investigate gravity–capillary waves resonantly excited by two topographic obstacles in a shallow water channel. By considering the weakly non-linear regime the forced fifth-order Korteweg–de Vries equation arises as a model for the free-surface displacement. The water surface is initially taken at rest over a uniform flow and the initial value problem for this equation is computed numerically using a pseudospectral method. We study nearly-resonant flows with intermediate capillary effects. Details of the wave interactions are analysed for obstacles with different sizes. Our numerical results indicate that the flow is not necessarily governed by the larger obstacle.

Integrability, bilinearization, solitons and exact three wave solutions for a forced Korteweg-de Vries equation

The Korteweg-de Vries (KdV) equation with a forcing term is considered which demonstrates the mathematical modelling of the physics of a shallow layer of fluid due to external forcing. The characteristic of integrability is investigated via Painlevè analysis and an auto Bäcklund transformation is fabricated via the truncated Painlevè expansion. Using the concept of Bell polynomial approach the bilinear Bäcklund transformation, Lax pair and infinite sequence of conservation laws are constructed. The one-soliton, two-soliton and three-soliton solutions are obtained via simplified Hirota method. We demonstrate the propagation and interaction of soliton solutions graphically. Moreover, a new form of ansatz in the exact three wave method (ETW) along with symbolic computation is applied and some new periodic type of three wave solutions which includes periodic two solitary wave solution, doubly periodic solitary wave solution and breather type of two solitary wave solution are obtained for the forced KdV equation.

Dynamical study of nonlinear ion acoustic waves in presence of charged space debris at Low Earth Orbital (LEO) plasma region

We consider the nonlinear ion acoustic wave induced by orbiting charged space debris objects in the plasma environment generated at the Low Earth Orbital (LEO) region. The generated nonlinear ion acoustic wave is shown to be governed by a forced Korteweg-de Vries (fKdV) equation with the forcing function dependent on the charged space debris function. For a specific relationship between the forcing debris function and the nonlinear ion acoustic wave, the forced KdV equation turns out to be a completely integrable system; where the debris function obeys a definite nonholonomic constraint. A special exact accelerated soliton solution (velocity of the soliton changes over time whereas its amplitude remains constant) has been derived for the ion acoustic wave for the first time. On the other hand, the amplitude of the solitonic debris function varies with time, and its shape changes during the propagation. Approximate ion acoustic solitary wave solutions with time varying amplitudes and velocities have been derived for different types of weakly localized charged debris functions. Possible applications of the obtained results in space plasma physics are stated along with future directions for research.

Three-Dimensional Kadomtsev–Petviashvili Damped Forced Ion Acoustic Solitary Waves from Orbital Debris

Subcentimeter debris has been proven to cause significant damage to Earth-orbiting spacecraft sensors and subsystems; however, they are currently undetectable using ground-based radar and optical methods. Orbital debris will produce plasma density solitary waves, or solitons, due to their electrical charge, which propagate along the debris velocity vector, with amplitudes within the range of detectability by existing ground- and space-based sensors. Previously, using the damped forced Korteweg–de Vries equation, the orbital altitudes, latitudes, and velocities, where solitons can be produced by millimeter- to centimeter-sized debris, were identified in the presence of Landau damping and considering one spatial dimension. The soliton amplitude, width, speed, wavelength, and propagation distance were also modeled as a function of debris size and speed. Now, the full 3-D signature of orbital debris solitons is simulated using the damped forced Kadomtsev–Petviashvili equation. Transverse solitonic perturbations extend across the width of the debris, with predictable amplitudes and speeds that can be approximated by the 1-D damped forced Korteweg–de Vries equation at the transverse soliton location. The transverse perturbations form soliton rings that advance ahead of the debris in the 3-D simulations, allowing for additional opportunity for detection. The presented 3-D soliton predictions will facilitate design of future detection methods. Orbital debris solitons would enable the first collision-free mapping of the small debris population.

A forced Korteweg–de Vries model for nonlinear mixing of oscillations in a dusty plasma

Nonlinear mixing of oscillations in a dusty plasma due to the harmonic time varying modulation of a nonlinear compressional oscillation is analyzed using a simple mathematical model consisting of a forced Korteweg–de Vries equation. An exact analytical solution of this equation is found to exhibit nonlinear mixing in the system. The model solution can be usefully employed to predict the existence of nonlinear mixing of oscillations in a two-dimensional dusty plasma system of a particular experimental configuration.

DEVELOPMENT OF RIGOROUS METHODS IN FLUID MECHANICS AND THEORY OF WATER WAVES

The thesis consists of five chapters where the following problems are considered - the transformation of long linear waves in an ocean with a variable depth; long wave scattering in a canal with a rapidly varying cross-section; long linear surface waves on stationary currents in a canal of constant depth and variable width; and a resonant interaction of a solitary wave with external pulse-type perturbations within the framework of forced Korteweg-de Vries equation. Chapter 1 reviews the history of these problems, and notes previous literature and research in this area. In Chapter 2, the transformation of long linear waves in an ocean of a variable depth is studied. The transformation coeffcients are considered as functions of frequency and the total depth drop for three typical models of bottom profile variation: piecewise linear, piecewise quadratic and hyperbolic tangent profiles. The results obtained for all these profiles are analysed and compared from the physical point of view. In Chapter 3, long wave scattering in a canal with a rapidly varying cross-section is studied. The scattering coeffcients are calculated for all possible incident wave orientations (background current downstream and upstream with respect to the background flow); and current types subcritical, transcritical, and supercritical. It is shown that when the over-reflection or over-transmission occurs, negative energy waves can appear in the flow. A spontaneous wave generation can happen in a transcritical accelerating flow, resembling a spontaneous wave generation on the horizon of an evaporating black hole due to the Hawking effect in astrophysics. In Chapter 4, long wave transformation is studied in the canal with abrupt variation of width and depth. Again, all possible wave orientation with respect to the background current is considered, and all types of a current is studied (subcritical, transcritical, and supercritical). The transformation coefficients are calculated and analysed as functions of wave frequency, Froud number, and depth drop. In Chapter 5, we revise the solutions of the forced Korteweg{de Vries equation for the resonant interaction of a solitary wave with the various external pulse-type perturbations. In contrast to the previous studies, we consider an arbitrary relationship between the width of a solitary wave and external forcing. In many cases, exact solutions of the forced Korteweg{de Vries equation can be obtained for the specific forcings of arbitrary amplitude. The theoretical outcomes obtained by asymptotic method are in good agreement with the results of direct numerical modelling within the framework of forced Korteweg-de Vries equation. In Conclusion, we summarise the results obtained within the various models and equations; discuss the future applications and innovation of the results; and suggest further perspectives for the future research.

Transcritical flow over obstacles and holes: Forced extended Korteweg–de Vries framework

A forced extended Korteweg-de Vries equation is used to simulate the transcritical flow of a density stratified fluid over localized topography. The extension of the usual Korteweg-de Vries equation is by the addition of a cubic nonlinear term as well as the usual quadratic nonlinear term, which is motivated by certain density stratifications, such as a two-layer fluid model. The case of negative forcing corresponds to a near-surface sharp pycnocline over an obstacle.

Simulating Damped Ion Acoustic Solitary Waves from Orbital Debris

Nontrackable orbital debris has become a great risk to Earth-orbiting spacecraft. Subcentimeter debris is currently undetectable using ground-based radar and optical methods. Recent analytical, computational, and experimental work has shown that plasma density solitary waves, or solitons, will be produced by orbital debris due to their electrical charge, and propagate along the debris velocity vector. The amplitude and velocity of solitons that may be produced by millimeter–centimeter-scale orbital debris, as well as the orbital altitudes, latitudes, and velocities where solitons can be produced as a function of debris size, have been previously described. The Chan–Kerkhoven pseudospectral method is used now to apply the damped forced Korteweg–de Vries equation, calculate the damping rate of the solitons, and estimate the resulting soliton propagation distance. It is demonstrated that Landau damping dominates over collisional damping for these solitons. Pinned solitons, which travel with the debris, do not experience significant damping due to the presence of the persistent debris force at the soliton location, and will continue to propagate until the debris enters a region where pinned solitons cannot be created. The median propagation distance for precursor solitons, which advance ahead of the debris, is 10 km for 0.5-mm-radius debris and 6 km for 0.5-cm-radius debris. With the current absence of a dedicated, calibrated, on-orbit debris detection sensor, plasma soliton detection would be the first collision-free method of mapping the small debris population. It is necessary to understand the damping of solitons in order to assess the feasibility of on-orbit debris detection.

Solution for fractional forced KdV equation using fractional natural decomposition method

The fractional natural decomposition method (FNDM) is employed in the present investigation to find the solution for fractional forced Korteweg-de Vries (FF-KdV) equation. Three distinct cases are chosen for each equation to validate and illustrate the effectiveness of the future technique. The behaviour for different values of Froude number (Fr) has been presented to assure the proficiency and reliability and of the considered method. Moreover, we captured the behaviour of the FNDM solution for distinct arbitrary order. The obtained results elucidate that, the considered method is very effective and easy to employ while analyse the behaviour of nonlinear fractional differential equations arising in connected areas of science and technology.

Transcritical flow over obstacles and holes: forced Korteweg–de Vries framework

This paper extends a previous study of free-surface flow over two localised obstacles using the framework of the forced Korteweg–de Vries equation, to an analogous study of flow over two localised holes, or a combination of an obstacle and a hole. Importantly the terminology obstacle or hole can be reversed for a stratified fluid and refers more precisely to the relative polarity of the forcing and the solitary wave solution of the unforced Korteweg–de Vries equation. As in the previous study, our main concern is with the transcritical regime when the oncoming flow has a Froude number close to unity. In the transcritical regime at early times, undular bores are produced upstream and downstream of each forcing site. We then describe the interaction of these undular bores between the forcing sites, and the outcome at very large times.

Rotational waves generated by current‐topography interaction

AbstractWe study nonlinear free‐surface rotational waves generated through the interaction of a vertically sheared current with a topography. Equivalently, the waves may be generated by a pressure distribution along the free surface. A forced Korteweg–de Vries equation (fKdV) is deduced incorporating these features. The weakly nonlinear, weakly dispersive reduced model is valid for small amplitude topographies. To study the effect of gradually increasing the topography amplitude, the free surface Euler equations are formulated in the presence of a variable depth and a sheared current of constant vorticity. Under constant vorticity, the harmonic velocity component is formulated in a simplified canonical domain, through the use of a conformal mapping which flattens both the free surface as well as the bottom topography. Critical, supercritical, and subcritical Froude number regimes are considered, while the bottom amplitude is gradually increased in both the irrotational and rotational wave regimes. Solutions to the fKdV model are compared to those from the Euler equations. We show that for rotational waves the critical Froude number is shifted away from 1. New stationary solutions are found and their stability tested numerically.

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.