Korteweg-de Vries Model Research Articles

Numerical investigations of fully nonlinear water waves generated by moving bottom topography

This paper is concerned with propagation of water waves induced by moving bodies with uniform velocity on the bottom of a channel, a simple model for prescribed underwater landslides. The fluid is assumed to be inviscid and incompressible, and the flow, irrotational. We apply this model to a variety of test problems, and particular attention is paid to long-time dynamics of waves induced by two landslide bodies moving with the same speed. We focus on the transcritical regime where the linear theory fails to depict the wave phenomena even in the qualitative sense since it predicts an infinite growth in amplitude. In order to resolve this problem, weakly nonlinear theory or direct numerical simulations for the fully nonlinear equations is required. Comparing results of the linear full-dispersion theory, the linear shallow water equations, the forced Korteweg-de Vries model, and the full Euler equations, we show that water waves generated by prescribed underwater landslides are characterized by the Froude number, sizes of landslide bodies and distance between them. Particularly, in the transcritical regime, the second body plays a key role in controlling the criticality for equal landslide bodies, while for unequal body heights, the higher one controls the criticality. The results obtained in the current paper complement numerical studies based on the forced Korteweg-de Vries equation and the nonlinear shallow water equations by Grimshaw and Maleewong (J. Fluid Mech. 2015, 2016).

Combined effects of topography and bottom friction on shoaling internal solitary waves in the South China Sea

A numerical study to a generalized Korteweg-de Vries (KdV) equation is adopted to model the propagation and disintegration of large-amplitude internal solitary waves (ISWs) in the South China Sea (SCS). Based on theoretical analysis and in situ measurements, the drag coefficient of the Chezy friction is regarded as inversely proportional to the initial amplitude of an ISW, rather than a constant as assumed in the previous studies. Numerical simulations of ISWs propagating from a deep basin to a continental shelf are performed with the generalized KdV model. It is found that the depression waves are disintegrated into several solitons on the continental shelf due to the variable topography. It turns out that the amplitude of the leading ISW reaches a maximum at the shelf break, which is consistent with the field observation in the SCS. Moreover, a dimensionless parameter defining the relative importance of the variable topography and friction is presented.

Experimental observation of cnoidal waveform of nonlinear dust acoustic waves

The experimentally measured waveform of nonlinear dust acoustic waves in a plasma is shown to be accurately described by a cnoidal function. This function, which is predicted by nonlinear theory, has broad minima and narrow peaks. Fitting the experimental waveforms to the cnoidal function also provides a measure of the wave's nonlinearity, namely, the elliptical parameter k. By characterizing experimental results at various wave amplitudes, we confirm that the parameter k increases and approaches a maximum value of unity, as the wave amplitude is increased. The underlying theory that predicts the cnoidal waveform as an exact solution of a Korteweg-de Vries model equation takes account of the streaming ions that are responsible for the spontaneous excitation of the dust acoustic waves.

Generation of dispersive shock waves in nonextensive plasmas

In this research we use a generalized hydrodynamic model to numerically investigate the quasi-neutral expansion and compression of an electron–ion plasma with nonextensive electron velocity distribution into or from vacuum. We study the effects of kinematic viscosity and electron–ion collisions on the expansion profile and compare our results to the numerical solutions of the standard Korteweg – de Vries (KdV) equation. It is found that the quasi-neutrality assumption in the hydrodynamic approach in the absence of viscosity and collisions, which leads to elimination of Poisson’s equation, sets the weak dispersion limit and becomes equivalent to the standard weakly dispersive KdV model. In the weak dispersion limit our model, as well as the KdV with small dispersion effect, predicts that a pulse-like initial profile evolves into solitary wave train. We further show that in a plasma expansion different shock profiles, such as purely dispersive, diffusive–dispersive, and dissipative ones, with significantly different characters may form. Finally, the effect of electron nonextensivity on oscillatory shock waves shows that the expansion profile is affected by changes of q-parameter. Our numerical solution is in qualitative agreement with some distinguished experiments showing the possibility of dispersive shock wave formation in rarefied plasma during an expansion into vacuum.

Classical symmetry analysis and exact solutions for generalized Korteweg–de Vries models with variable coefficients

In this paper, by using the classical symmetry analysis method symmetries for the generalized variable-coefficient Korteweg–de Vries model are obtained. Then, the reduced nonlinear ordinary differential equations with variable coefficients are solved by auxiliary equation method. Hermite differential equation is chosen as an auxiliary equation and some new exact solutions for the nonlinear partial differential equation in hand are obtained.

On the solutions and conservation laws of a two-dimensional Korteweg de Vries model: Multiple exp-function method

Abstract Under investigation in this paper is a two-dimensional Korteweg de Vries model, which is a spacial extension of the Korteweg de Vries model. An infinite number of nonlocal conservation laws are given which indicate the integrability of this model. Exact soliton solutions are then respectively derived by means of the multiple exp-function method.

Numerical study on the quantitative error of the Korteweg–de Vries equation for modelling random waves on large scale in shallow water

The Korteweg–de Vries (KdV) equation is often adopted to simulate phase-resolved random waves on large scale in shallow water. It shows that the KdV equation is computationally efficient and can give sufficiently accurate results, but it is not always suitable and the error by using it cannot be predicted. This paper attempts to give the quantitative formulas for estimating the error of the statistics when simulating random waves in shallow water by using it. The formulas are obtained by fitting the errors of the KdV equation in comparison with the fully nonlinear model using the same initial condition based on the Wallops spectrum with a wide range of parameters. This paper also demonstrates how the formulas would be used, e.g., to estimate the error of the results by using the KdV model, or to justify its suitability for modelling random waves on large scale in shallow water.

Jacobi elliptic wave solutions for two variable coefficients cylindrical Korteweg–de Vries models arising in dusty plasmas by using direct reduction method

In this paper, generalized models for both (2+1)-dimensional cylindrical modified Korteweg–de Vries (cmKdV) equation with variable coefficients and (3+1)-dimensional variable coefficients cylindrical Korteweg–de Vries (cKdV) equation are studied by direct reduction method. A direct reduction to nonlinear ordinary differential equations in the form of Riccati equations obtained for the considered equations under some integrability conditions. The search for solutions for the reduced Riccati equations has yielded many Jacobi elliptic wave solutions, solitary and periodic wave solutions for both (2+1)-dimensional cmKdV and (3+1)-dimensional cKdV equations. Physical application for the obtained solutions as dust ion acoustic waves in plasma physics is given

Simulations and observation of nonlinear internal waves on the continental shelf: Korteweg–de Vries and extended Korteweg–de Vries solutions

Abstract. Numerical solutions of the Korteweg–de Vries (KdV) and extended Korteweg–de Vries (eKdV) equations are used to model the transformation of a sinusoidal internal tide as it propagates across the continental shelf. The ocean is idealized as being a two-layer fluid, justified by the fact that most of the oceanic internal wave signal is contained in the gravest mode. The model accounts for nonlinear and dispersive effects but neglects friction, rotation and mean shear. The KdV model is run for a number of idealized stratifications and unique realistic topographies to study the role of the nonlinear and dispersive effects. In all model solutions the internal tide steepens forming a sharp front from which a packet of nonlinear solitary-like waves evolve. Comparisons between KdV and eKdV solutions are made. The model results for realistic topography and stratification are compared with observations made at moorings off Massachusetts in the Middle Atlantic Bight. Some features of the observations compare well with the model. The leading face of the internal tide steepens to form a shock-like front, while nonlinear high-frequency waves evolve shortly after the appearance of the jump. Although not rank ordered, the wave of maximum amplitude is always close to the jump. Some features of the observations are not found in the model. Nonlinear waves can be very widely spaced and persist over a tidal period.

High Order $$C^0$$ C 0 -Continuous Galerkin Schemes for High Order PDEs, Conservation of Quadratic Invariants and Application to the Korteweg-de Vries Model

We address the Korteweg-de Vries equation as an interesting model of high order partial differential equation, and show that it is possible to develop reliable and effective schemes, in terms of accuracy, computational efficiency, simplicity of implementation and, if required, conservation of the lower invariants, on the basis of a (only) $$H^1$$ -conformal Galerkin approximation, namely the Spectral Element Method. The proposed approach is a priori easily extensible to other partial differential equations and to multidimensional problems.

A weak-constraint 4DEnsembleVar. Part II: experiments with larger models

ABSTRACTIn recent years, hybrid data-assimilation methods which avoid computation of tangent linear and adjoint models by using ensemble 4-dimensional cross-time covariances have become a popular topic in Numerical Weather Prediction. 4DEnsembleVar is one such method. In spite of its capabilities, its application can sometimes become problematic due to the not-trivial task of localising cross-time covariances. In this work we propose a formulation that helps to alleviate such issues by exploiting the presence of model error, i.e. a weak-constraint 4DEnsembleVar. We compare the weak-constraint 4DEnsembleVar to that of other data-assimilation methods. This is part II of a two-part paper. In part I, we describe the 4DEnsembleVar framework and problems with localised temporal cross-covariances associated with this method are discussed and illustrated on the Korteweg de Vries model. We also introduce our weak-constraint 4DEnsemble-Var formulation and show how it can alleviate—at least partially—the problem of having low-quality time cross-covariances. The second part of this paper deals with experiments on larger and more complicated models, namely the Lorenz 1996 model and a modified shallow-water model with simulated convection, both of them under the presence of model error. We investigate the performance of weak-constraint 4DEnsembleVar against strong-constraint 4DEnsembleVar (both with and without localisation) and other traditional methods (4DVar and the Local Ensemble Transform Kalman Smoother). Using the analysis root mean square error (RMSE) as a metric, these methods have been compared considering observation density (in time and space), observation period, ensemble sizes and assimilation window length. In this part we also explain how to perform outer loops in the EnVar methods. We show that their use can be counter-productive if the presence of model error is ignored by the assimilation method. We show that the addition of a weak-constraint generally improves the RMSE of 4DEnVar in cases where model error has time to develop, especially in cases with long assimilation windows and infrequent observations. We have assumed good knowledge of the statistics of this model error.

Obliquely propagating dust ion-acoustic solitary waves and double layers in multicomponent plasmas

This research work deals with the propagation characteristics of dust ion-acoustic solitary waves and double layers in a strongly magnetized and rotating plasma comprising of fluid ions, charged dust, superthermal electrons, and positrons. In small amplitude approximation, reductive perturbation technique is employed to derive the Korteweg-de Vries (KdV) equation, and its analytical solution is presented. The combined effects of variation of different plasma parameters like superthermality, dust concentration, magnetic field strength, and rotation of a plasma on the amplitude and width of dust ion-acoustic solitons are analyzed. Both positive and negative potential solitary waves occur in this dusty plasma system. The critical values of plasma parameters for which KdV model is not valid, are examined, and the modified KdV (mKdV) equation is derived. The existence regimes of mKdV solitons and double layers have also been investigated. Positive and negative potential double layers occur in the present plasma system.

Internal solitary waves in a two-fluid system with a free surface

Internal solitary waves in a system of two fluids, silicone oil and water, bounded above by a free surface are studied both experimentally and theoretically. By adjusting an extra volume of silicone oil released from a reservoir, a wide range of amplitude waves are generated in a wave tank. Wave profiles as well as wave speeds are measured using multiple wave probes and are then compared with both the weakly nonlinear Korteweg–de Vries (KdV) models and the strongly nonlinear Miyata–Choi–Camassa (MCC) models. As the density difference between the two fluids in the experiment is relatively small (approximately 14 %), but non-negligible, special attention is paid to the effect of the boundary condition at the top surface. The nonlinear models valid for rigid-lid (RL) and free-surface (FS) boundary conditions are considered separately. It is found that the solitary wave of the FS model for a given amplitude is consistently narrower than that of the RL model and it propagates at a slightly lower speed. Due to strong nonlinearity in the internal-wave motion, the weakly nonlinear KdV models fail to describe the measured internal solitary wave profiles of intermediate and large wave amplitudes. The strongly nonlinear MCC-FS model agrees better with the measurements than the MCC-RL model, which indicates that the free-surface boundary condition at the top surface is crucial in describing the internal solitary waves in the experiment correctly. Leaving the top surface free in the experiment allows us to observe small and relatively short wave packets on the top surface, particularly when the amplitude of the internal solitary wave is large. Once excited, the wave packet is located above the front half of the internal solitary wave and propagates with a speed close to that of the internal solitary wave underneath. A simple resonance mechanism between short surface waves and long internal waves without and with nonlinear effects is examined to estimate the characteristic wavelength of modulated short surface waves, which is found to be in good agreement with the observed wavelength when nonlinearity is taken into account. Using ray theory, the evolution of short surface waves in the presence of a background current induced by an internal solitary wave is also investigated to examine the location of the modulated surface wave packet.

Experimental observation of precursor solitons in a flowing complex plasma.

The excitation of precursor solitons ahead of a rapidly moving object in a fluid, a spectacular phenomenon in hydrodynamics that has often been observed ahead of moving ships, has surprisingly not been investigated in plasmas where the fluid model holds good for low frequency excitations such as ion acoustic waves. In this Rapid Communication we report an experimental observation of precursor solitons in a flowing dusty plasma. The nonlinear solitary dust acoustic waves (DAWs) are excited by a supersonic mass flow of the dust particles over an electrostatic potential hill. In a frame where the fluid is stationary and the hill is moving the solitons propagate in the upstream direction as precursors while wake structures consisting of linear DAWs are seen to propagate in the downstream region. A theoretical explanation of these excitations based on the forced Korteweg-deVries model equation is provided and their practical implications in situations involving a charged object moving in a plasma are discussed.

Hybrid (Vlasov-Fluid) simulation of ion-acoustic solitons chain formation including trapped electrons

Disintegration of a Gaussian profile into ion-acoustic solitons in the presence of trapped electrons [H. Hakimi Pajouh and H. Abbasi, Phys. Plasmas 15, 082105 (2008)] is revisited. Through a hybrid (Vlasov-Fluid) model, the restrictions associated with the simple modified Korteweg de-Vries (mKdV) model are studied. For instance, the lack of vital information in the phase space associated with the evolution of electron velocity distribution, the perturbative nature of mKdV model which limits it to the weak nonlinear cases, and the special spatio-temporal scaling based on which the mKdV is derived. Remarkable differences between the results of the two models lead us to conclude that the mKdV model can only monitor the general aspects of the dynamics, and the precise picture including the correct spatio-temporal scales and the properties of solitons should be studied within the framework of hybrid model.

Analytical study on a two-dimensional Korteweg–de Vries model with bilinear representation, Bäcklund transformation and soliton solutions

With symbolic computation, Bell-polynomial scheme and bilinear method are applied to a two-dimensional Korteweg–de Vries (KdV) model, which is firstly proposed with Lax pair generating technique. Bell-polynomial expression with one auxiliary independent variable is derived and transformed into bilinear form. According to the coupled two-field conditions between the primary and replica fields, Bell-polynomial-typed Bäcklund transformations (BTs) are constructed and converted into the bilinear ones. Finally, soliton solutions of the two-dimensional KdV model are obtained (via solving the bilinear representation and BT, respectively) and compared. Such associated integrable properties as bilinear representation, BT (especially auxiliary-independent-variable-involved Bell-polynomial-typed ones constructed in this paper) and soliton solutions (especially the multi-soliton ones) may be useful for further study on other two-dimensional KdV and KdV-typed models.

Stability of steady gravity waves generated by a moving localised pressure disturbance in water of finite depth

We study the stability of the steady waves forced by a moving localised pressure disturbance in water of finite depth. The steady waves take the form of a downstream wavetrain for subcritical flow, but for supercritical flow there is a solution branch for each specified forcing term, which has two localised solitary-like solutions for each Froude number, a small-amplitude and a large-amplitude solution. Our main purpose is to numerically investigate the stability of these steady waves by simulating the unsteady development from appropriate initial conditions. We use a forced Korteweg-de Vries model valid in the transcritical regime for weak forcing, and a fully nonlinear boundary integral simulation. The simulations using the forced Korteweg-de Vries model are in good agreement with the fully nonlinear simulations when the Froude number is near unity for small pressure forcing, as expected. We find that the steady subcritical downstream wavetrain is stable. For supercritical flow, the small-amplitude solitary-like wave which has bifurcated from a uniform flow is stable, whereas the large-amplitude solitary-like wave which has bifurcated from a free solitary wave is unstable. Furthermore, here a difference between the forced Korteweg-de Vries model and the fully nonlinear simulations is revealed. For moderate and large pressure forcing, the forced Korteweg-de Vries model predicts a stable solitary wave moving away from the pressure forcing, while the nonlinear simulation shows that this wave evolves to a breaking wave.

Wave-making experiments and theoretical models for internal solitary waves in a two-layer fluid of finite depth

A laboratory wave-making method is developed for the internal solitary wave under the condition of giving its amplitude produced by oppositely and horizontally pushing two vertical plates placed separately in the upper- and lower-layer fluids of a large-scale density stratified tank where based on the Miyata-Choi-Camassa (MCC) theoretical model, the layer-mean velocities of the upper- and lower-layer fluids induced by the internal solitary wave are used as the velocities of the two plates. On this basis, a series of experiments is conducted to explore the applicability conditions for internal solitary wave theories with stationary solutions which are Korteweg-de Vries (KdV), extended KdV (eKdV), MCC and modified KdV (mKdV) models in a two-layer fluid of finite depth respectively. It is shown that for the nonlinear parameter ε and the dispersion parameter μ defined by the total water depth, there exists a critical dispersion parameter μ0, in the case of μ μ0, the KdV model is applicable for ε ≤μ, the eKdV model is applicable for μ ε ≤√μ, as well as the MCC model is applicable for ε > √μ. However, in the case of μ ≥ μ0, the MCC model is still applicable for a wide range of ε. Furthermore, for the case where the ratio of depth between the upper- and lower-layer fluids is not close to its critical value, the mKdV model is mainly applicable for the case where the amplitude of the internal solitary wave is close to its theoretical limiting amplitude, however, the MCC model is also applicable for such a case. The investigation quantitatively characterizes the applicability conditions for four classes of internal solitary wave theories, and provides an important theoretical foundation for what kinds of theories can be chosen to model internal solitary waves in the ocean.

Derivation of a modified Korteweg–de Vries model for few-optical-cycles soliton propagation from a general Hamiltonian

Propagation of few-cycles optical pulses in a centrosymmetric nonlinear optical Kerr (cubic) type material described by a general Hamiltonian of multilevel atoms is considered. Assuming that all transition frequencies of the nonlinear medium are well above the typical wave frequency, we use a long-wave approximation to derive an approximate evolution model of modified Korteweg–de Vries type. The model derived by rigorous application of the reductive perturbation formalism allows one the adequate description of propagation of ultrashort (few-cycles long) solitons.

THE GENERALIZED WRONSKIAN SOLUTIONS OF THE INTEGRABLE VARIABLE-COEFFICIENT KORTEWEG-DE VRIES EQUATION

A broad set of sufficient conditions consisting of systems of linear partial differential equations are presented which guarantee that the Wronskian determinant is the solutions of the integrable variable-coefficient Korteweg-de Vries model from Bose–Einstein condensates and fluid dynamics. The generalized Wronskian solutions provide us with a comprehensive approach to construct many exact solutions including rational solutions, solitons, negatons, positons, and complexitons.