Nonlinear analysis and solitary waves for two superposed streaming electrified fluids of uniform depths with rigid boundaries

The nonlinear modulation of the interfacial waves of two superposed dielectric fluids with uniform depths and rigid horizontal boundaries, under the influence of constant normal electric fields and uniform horizontal velocities, is investigated using the multiple-time scales method. It is found that the behavior of small perturbations superimposed on traveling wave trains can be described by a nonlinear Schrödinger equation in a frame of reference moving with the group velocity. Wave-like solutions to this equation are examined, and different types of localized excitations (envelope solitary waves) are shown to exist. It is shown that when these perturbations are neutrally stable and sufficiently long, solutions to the nonlinear Schrödinger equation may be approximated by the well-known Korteweg-de Vries equation. The speed of the solitary on the interface is seen to be reduced by the electric field. It is found that there are two critical values of the applied voltage that lead to (i) breaking up of the solitary waves, and (ii) bifurcation of solutions of the governing equations. On the other hand, the complex amplitude of standing wave trains near the marginal state is governed by a similar type of nonlinear Schrödinger equation in which the roles of time and space are interchanged. This equation, under a suitable transformation, is obtained as the Korteweg-de Vries equation with a variable coefficient. It is shown that this type of equations admit a solitary wave type of solutions with variable speed. Using the tangent hyperbolic method, it is observed that the wave speed increases as well as decreases, with the increase of electric field values, according to the chosen wavenumbers range. Finally, the nonlinear stability analysis is discussed in view of the coefficients of nonlinear Schrödinger equation to show the effects of various physical parameters, and also to recover the some limiting cases studied earlier in the literature.     

Well-Posedness for Multicomponent Schrödinger–gKdV Systems and Stability of Solitary Waves with Prescribed Mass

In this paper we prove the well-posedness issues of the associated initial value problem, the existence of nontrivial solutions with prescribed \(L^2\)-norm, and the stability of associated solitary waves for two classes of coupled nonlinear dispersive equations. The first problem here describes the nonlinear interaction between two Schrödinger type short waves and a generalized Korteweg-de Vries type long wave and the second problem describes the nonlinear interaction of two generalized Korteweg-de Vries type long waves with a common Schrödinger type short wave. The results here extend many of the previously obtained results for two-component coupled Schrödinger–Korteweg-de Vries systems.     

Damping of large-amplitude solitary waves

We consider the damping of large-amplitude solitary waves in the framework of the extended Korteweg-de Vries equation (that is, the usual Korteweg-de Vries equation supplemented with a cubic nonlinear term) modified by the inclusion of a small damping term. The damping of a solitary wave is studied for several different forms of friction, using both the analytical adiabatic asymptotic theory and numerical simulations. When the coefficient of the cubic nonlinear term has the opposite sign to the coefficient of the linear dispersive term, the extended Kortweg-de Vries equation can support large-amplitude “thick” solitary waves. Under the influence of friction, these “thick” solitary waves decay and may produce one or more secondary solitary waves in this process. On the other hand, when the coefficient of the cubic nonlinear term has the same sign as the coefficient of the linear dispersive term, but the opposite sign to the coefficient of the quadratic nonlinear term, the action of friction may cause a solitary wave to decay into a wave packet.     

Particle trajectories under interactions between solitary waves and a linear shear current

This paper is concerned with particle trajectories beneath solitary waves when a linear shear current exists. The fluid is assumed to be incompressible and inviscid, lying on a flat bed. Classical asymptotic expansion is used to obtain a Korteweg-de Vries(Kd V) equation, then a forth-order Runge-Kutta method is applied to get the approximate particle trajectories. On the other hand, our particular attention is paid to the direct numerical simulation(DNS) to the original Euler equations. A conformal map is used to solve the nonlinear boundary value problem. Highaccuracy numerical solutions are then obtained through the fast Fourier transform(FFT) and compared with the asymptotic solutions, which shows a good agreement when wave amplitude is small. Further, it also yields that there are different types of particle trajectories. Most surprisingly,periodic motion of particles could exist under solitary waves, which is due to the wave-current interaction.     

The evolution and interaction of Marangoni-Bénard solitary waves

Marangoni-Bénard convection is the process by which oscillatory waves are generated on an interface due to a change in surface tension. This process, which can be mass or temperature driven is described by a perturbed Korteweg-de Vries (KdV) equation. The evolution and interaction of solitary waves generated by Marangoni-Bénard convection is examined. The solitary wave with steady-state amplitude, which occurs when the excitation and friction terms of the perturbed KdV equation are in balance is found to second-order in the perturbation parameter. This solitary wave has a fixed amplitude, which depends on the coefficients of the perturbation terms in the governing equation. The evolution of a solitary wave of arbitrary amplitude to the steady-state amplitude is also found, to first-order in the perturbation parameter. In addition, by using a perturbation method based on inverse scattering, it is shown that the interaction of two solitary waves is not elastic with the change in wave amplitude determined. Numerical solutions of the perturbed KdV equation are presented and compared to the asymptotic solutions.     

Solitary waves in stratified fluids and their interaction

A systematic procedure is proposed for obtaining solutions for solitary waves in stratified fluids. The stratification of the fluid is assumed to be exponential or linear. Its comparison with existing results for an exponentially stratified fluid shows agreement, and it is found that for the odd series of solutions the direction of displacement of the streamlines from their asymptotic levels is reversed when the stratification is changed from exponential to linear. Finally the interaction of solitary waves is considered, and the Korteweg-de Vries equation and the Boussinesq equation are derived. Thus the known solutions of these equations can be relied upon to provide the answers to the interaction problem.     

Adjustment processes and radiating solitary waves in a regularised ostrovsky equation

The Ostrovsky equation is an adaptation of the Korteweg-de Vries equation widely used to describe the effect of rotation on surface and internal solitary waves. It has been shown that the effect of rotation is to destroy such solitary waves in finite time due to the emission of trailing radiation. Here this issue is re-examined for a regularised Ostrovsky equation. The regularisation is necessary to remove an anomaly in the Ostrovsky equation whereby there is a discontinuity in the mass field at the initial moment. It is demonstrated that in the regularised Ostrovsky equation there is a rapid adjustment of the mass which is transported a large distance in the opposite direction to that in which the solitary wave propagates.     

Interactions among different types of nonlinear waves described by the Kadomtsev–Petviashvili equation

In nonlinear science, the interactions among solitons are well studied because the multiple soliton solutions can be obtained by various effective methods. However, it is very difficult to study interactions among different types of nonlinear waves such as the solitons (or solitary waves), the cnoidal periodic waves and Painlevé waves. In this paper, taking the Kadomtsev–Petviashvili (KP) equation as an illustration model, a new method is established to find interactions among different types of nonlinear waves. The nonlocal symmetries related to the Darboux transformation (DT) of the KP equation is localized after embedding the original system to an enlarged one. Then the DT is used to find the corresponding group invariant solutions. It is shown that the essential and unique role of the DT is to add an additional soliton on a Boussinesq-type wave or a KdV-type wave, which are two basic reductions of the KP equation.     

分层流体中gKdV型孤立波的迎撞

本文采用约化摄动法和PLK方法并通过双参数摄动展开,讨论了分层流体中以推广的Korteweg-de vries方程(gKdV方程)描述的孤立波的迎撞问题,求得了二阶近似解。分析结果表明,gKdV型孤立波碰撞后保持原来的形状不变,在碰撞时最大波幅为两个来碰孤立波的最大波幅的线性叠加。     

Nonlinear dynamical magnetosonic wave interactions and collisions in magnetized plasma

The one-dimensional nonlinear dynamical wave interactions in a system of quasineutral two-fluid plasma in a constant magnetic field are investigated.The existence of the travelling wave solutions is discussed.The modulation stability of linear waves and the modulation instability of weakly nonlinear waves are presented.Both suggest that the Korteweg-de Vries(KdV) system is modulationally stable.Besides,the wave interactions including the periodic wave interaction and the solitary wave interaction are captured and presented.It is shown that these interacting waves alternately exchange their energy during propagation.The Fourier spectrum analysis is used to depict the energy transformation between the primary and harmonic waves.It is known that the wave interactions in magnetized plasma play an important role in various processes of heating and energy transportation in space and astrophysical plasma.However,few researchers have considered such magnetohydrodynamic(MHD) wave interactions in plasma.It is expected that this work can provide additional insight into understanding of behaviors of MHD wave interactions.     

Problem of Breakdown of Solitary Waves and Discontinuities

The evolution of initial data of the solitary-wave type with time is investigated numerically. The solitary wave amplitude decreases due to the generation of short-wave radiation. This solution is interpreted as the solution with a discontinuity qualitatively analogous to the solution of the problem of the breakdown of an arbitrary discontinuity in dissipationless systems. The solitary wave amplitude reduction rate is estimated, first for a generalized Korteweg-de Vries equation and then for plasma waves. Features of the investigation are analyzed for cold and hot-electron plasmas.     

Stability of solitary waves in dispersive media described by a fifth-order evolution equation

The problem of the existence and dynamical stability of solitary wave solutions to a fifth-order evolution equation, generalizing the well-known Korteweg-de Vries equation, is treated. The theoretical framework of the paper is largely based on a recently developed version of positive operator theory in Fréchet spaces (which is used for the existence proof) and the theory of orbital stability for Hamiltonian systems with translationally invariant Hamiltonians. The validity of sufficient conditions for stability are established. The shape of solitary waves under analysis are determined by a numerical solution of the boundary-value problem followed by a correction using the Picard method of 4–12 orders of accuracy.     

Wave number shocks for the leading tail of Korteweg-de Vries solitary waves in slowly varying media

The leading tail for slowly varying solitary waves for the perturbed Korteweg-de Vries (KdV) equation is analyzed. The path of the core of the solitary wave is obtained and shown to provide a moving boundary for the leading tail. The leading tail is predicted to be triple valued within a penumbral caustic (envelope of characteristics) caused by the initial acceleration of the core. A rescaling in the neighborhood of the singularity shows that the solution there satisfies the diffusion equation. The solution involves an incomplete Airy-type exponential integral, where critical points (significant for Laplace's asymptotic method) satisfy the structure of the penumbral caustic. A wave number shock develops, which separates two different solitary wave tails, one due to the moving core and the other due to the initial condition. The shock velocity is that predicted from conservation of waves.     

Stability of the Korteweg-de Vries solitary wave solution

ＳＴＡＢＩＬＩＴＹＯＦＴＨＥＫＯＲＴＥＷＥＧ－ＤＥＶＲＩＥＳＳＯＬＩＴＡＲＹＷＡＶＥＳＯＬＵＴＩＯＮＬｕＸｉａｎ－ｑｉｎｇ（吕咸青）（ＳｈａｎｄｏｎｇＥｄｕｃａｔｉｏｎＣｏｌｌｅｇｅ,Ｊｉｎａｎ）（ＲｅｃｅｉｖｅｄＪｕｎｅ１８,１９９３,Ｃｏｍｍｕｎ...     

Waves in an inhomogeneous plasma with Hall dispersion

The evolution of steady-state periodic solutions of the Korteweg-de Vries equation (the socalled cnoidal waves), propagating along the direction of the gravitational force with an arbitrary orientation of the magnetic field, is studied for plasma characterized by Hall dispersion and Joule dissipation, using the magnetohydrodynamic approximation. The wavelength is regarded as much shorter than the characteristic scale of the inhomogeneity. The dependence of the wave amplitude on the distance to the source of the wave is considered for various limiting cases. The behavior of the wave depends on the temperature distribution in the medium. In the particular case of an isothermal atmosphere, the problem is solved analytically for a cold plasma in the absence of dissipation. The amplitude of both fast and slow waves increases when the wave travels upward and diminishes when the wave travels downward. The nonlinearity of the wave (i.e., the parameter characterizing the deviation of the wave from sinusoidal form) diminishes in the case of fast magnetoacoustic waves when the wave travels upward and increases when the wave travels downward. The situation is reversed for slow magnetoacoustic waves.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 139–144, September–October, 1976.The author is grateful to V. B. Baranov for constant interest in the work and valuable comments.     

Nonlinear waves on the free surface of an electrically conducting liquid

Gravity waves of small but finite amplitude on the free surface of an electrically conducting liquid are examined. For the waves whose propagation is described by the Korteweg-de Vries equation (in the absence of a magnetic field), equations are derived. In addition to nonlinearity and dispersion, these equations include the influence of applied magnetic fields. As an example, the effect of magnetic damping on a solitary wave is presented.     

The splitting of solitary waves running over a shallower water

The Korteweg-de Vries type of equation (called KdV-top) for uni-directional waves over a slowly varying bottom that has been derived by Van Groesen and Pudjaprasetya [E. van Groesen, S.R. Pudjaprasetya, Uni-directional waves over slowly varying bottom. Part I. Derivation of a KdV-type of equation, Wave Motion 18 (1993) 345–370.] is used to describe the splitting of solitary waves, running over shallower water, into two (or more) waves. Results of numerical computations with KdV-top are presented; qualitative and quantitative comparisons between the analytical and numerical results show a good agreement.     

Nonlinear resonances and wave jumps in media with higher-order dispersion

A new technique for systematically investigating biperiodic (two-wave) steady-state solutions is described with reference to modified Korteweg-de Vries and Schrödinger equations which generalize the conventional model equations for waves on water, in plasmas, and in nonlinear optics [1]. Among these solutions those with ordinary and resonance wave interactions are distinguished. Both singular solutions similar to the solitons of a resonantly interacting wave envelope and solitary waves are found. The soliton-like solutions obtained are used for describing the wave jump structure.Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 4, pp. 113–124, July–August, 1996.