Korteweg-de Vries Waves Research Articles

From decoupled integrable models to coupled ones via a deformation algorithm* *Supported by The National Natural Science Foundation (Nos. 12235007, 12090020, 11975131, 12090025).

By using a reconstruction procedure of conservation laws of different models, the deformation algorithm proposed by Lou, Hao and Jia has been used to a new application such that a decoupled system becomes a coupled one. Using the new application to some decoupled systems such as the decoupled dispersionless Korteweg–de Vries (KdV) systems related to dispersionless waves, the decoupled KdV systems related to dispersion waves, the decoupled KdV and Burgers systems related to the linear dispersion and diffusion effects, and the decoupled KdV and Harry–Dym (HD) systems related to the linear and nonlinear dispersion effects, we have obtained various new types of higher dimensional integrable coupled systems. The new models can be used to describe the interactions among different nonlinear waves and/or different effects including the dispersionless waves (dispersionless KdV waves), the linear dispersion waves (KdV waves), the nonlinear dispersion waves (HD waves) and the diffusion effect. The method can be applied to couple all different separated integrable models.

Particle motions beneath irrotational water waves

Neutral and buoyant particle motions in an irrotational flow are investigated under the passage of linear, nonlinear gravity, and weakly nonlinear solitary waves at a constant water depth. The developed numerical models for the particle trajectories in a non-turbulent flow incorporate particle momentum, size, and mass (i.e., inertial particles) under the influence of various surface waves such as Korteweg-de Vries waves which admit a three parameter family of periodic cnoidal wave solutions. We then formulate expressions of mass-transport velocities for the neutral and buoyant particles. A series of test cases suggests that the inertial particles possess a combined horizontal and vertical drifts from the locations of their release, with a fall velocity as a function of particle material properties, ambient flow, and wave parameters. The estimated solutions exhibit good agreement with previously explained particle behavior beneath progressive surface gravity waves. We further investigate the response of a neutrally buoyant water parcel trajectories in a rotating fluid when subjected to a series of wind and wave events. The results confirm the importance of the wave-induced Coriolis-Stokes force effect in both amplifying (destroying) the pre-existing inertial oscillations and in modulating the direction of the flow particles. Although this work has mainly focused on wave-current-particle interaction in the absence of turbulence stochastic forcing effects, the exercise of the suggested numerical models provides additional insights into the mechanisms of wave effects on the passive trajectories for both living and nonliving particles such as swimming trajectories of plankton in non-turbulent flows.

Higher-order nonlinearity and double layers in kinetic Alfvén waves and ion-acoustic waves with two temperature isothermal electrons

Using a new technique following the Bogoliubov Mitropolsky method, higher-order nonlinear and dispersive effects are obtained for a kinetic Alfven wave and ion-acoustic wave with two temperature isothermal electrons, starting from a set of equations that lead to a linear dispersion relation coupling kinetic Alfven wave ion-acoustic wave. Higher-order solitary wave profiles are obtained for both waves. Graphs are plotted showing the variation of both the Mach number and soliton width against the soliton amplitude for different angles of propagation with the applied magnetic field. Amplitude, width and velocity are found to be greater for larger angles of propagation for both waves. It is found that the higher-order ion-acoustic wave moves faster than the Korteweg-de Vries (KdV) wave, but the higher-order Alfven wave is slower. The higher-order amplitude and width of both waves are found to be less than those of KdV waves. The potential structure and widths of the double layer for both waves are shown graphically. It is found that the amplitude of the double layer is independent of the angles of propagation, but the widths are greater for larger angles of propagation for both waves.

Mass transport theory for the Toda lattices, dispersive and dissipative

To establish mass transport theory on nonlinear lattices, we formulate the Korteweg–deVries (KdV) equation and the Burgers equation using the flow variable representation so as to facilitate comparison with the Boltzmann equation and with the Cahn–Hilliard equation in classical statistical mechanics. We also study Toda lattice microdynamics using the Flaschka representation, and compare with the Liouville equation. Like the linear diffusion equation, the Boltzmann equation and the Liouville equation are to be solved for a distribution function, which is intrinsically probabilistic. Transport theory in linear systems is governed by the isotropic motions of the kinetic equations. In contrast, the KdV perturbation equation derived from the Toda lattice microdynamics expresses hydrodynamic mass transport. The KdV equation in hydrodynamics and the Burgers equation in thermodynamics do not involve a probability distribution function. The nonlinear lattices do not retain isotropy of the mass transport equations. In consequence, it is proposed that in the presence of hydrodynamic flows to the left, KdV wave propagation proceeds to the right. This basic property of the KdV system is extended to thermodynamics in the Burgers system. These features arise because linear systems are driven towards an equilibrium by molecular collisions, whereas the inhomogeneities of the nonlinear lattices are generated by the potential energy of interaction. Diffusion as expressed by the Burgers equation is governed not only by a chemical potential, but also by the Toda lattice potential energy.

Emergence and control of multiphase nonlinear waves by synchronization.

Large amplitude multiphase solutions of the periodic Korteweg-de Vries equation are excited and controlled by a small forcing. The approach uses passage through an ensemble of resonances and subsequent multiphase self-locking of the system with eikonal-type perturbations. The synchronization of each phase in the Korteweg-de Vries wave is robust, provided the corresponding driving amplitude exceeds a threshold.

On a class of internal solitary waves in a two-layer fluid

Solitary waves on an interface between two fluids are considered. A uniform asymptotic expansion is constructed for internal solitary waves with flat crests (of the plateau type) that degenerate into a bore in the limit. It is shown that, in this case, in contrast to a Korteweg-de Vries wave, the wave amplitude is of the same order of smallness as the longwave approximation parameter.

Long wave interaction over varying topography

The propagation of long waves on the surface of a three-dimensional fluid domain bounded below by slowly varying topography is considered. There are two important limits: If the initial data can be written in terms of a discrete set of one-dimensional wavefronts, the resulting wave field is described by a set of variable coefficient Korteweg-de Vries (KdV) equations for each wave along its characteristic curve. Waves along different characteristics interact with each other yielding phase shifts that depend on the wave amplitudes, the angle between the rays and the local depth. If the initial data is modulated slowly in the direction parallel to the wavefronts, the wave field is described by variable coefficient Kadomtsev-Petviashvili (KP) equations along rays. For topography varying only in one direction, we calculate explicit results for the interaction between two sets of periodic or solitary waves and show the equivalence of a single nearly normally incident KdV wave and a normally incident KP wave.

Two-level solitary waves as generalized solutions of the KdV equation

An analytical class of standing wave solutions to the Korteweg–de Vries (KdV) equation is obtained in the framework of continuous finite generalized functions. This paper shows how this class can be used to describe a great variety of wave shapes, especially bores and jumps. These new solutions are built by appropriately combining parts of two ordinary KdV waves. These represent a system with a steep transition between different energy levels of two potential wells. A number of specific cases of the generalized solutions are identical to those obtained recently within the theory of the KdV equation forced by a Dirac delta function. Numerical simulations of both stationary and transient KdV equations are carried out in a few cases. The weak formulation used in the numerical scheme is equivalent to the analytical generalized functions approach. Simulations of initially perturbed wave fronts prove the high degree of stability of many of these solutions.

New type of gap soliton in a coupled Korteweg-de Vries wave system.

We show that, in a narrow gap in the spectrum of two linearly coupled Korteweg--de Vries equations with opposite signs of the dispersion coefficient, a two-parameter family of solitons of a novel type may exist. These are envelope solitons with decaying oscillating tails, which are radically different from the gap solitons previously known in nonlinear optics. In particular, they may become singular at some value of the velocity, and degenerate into algebraic solitons in another special case. It is demonstrated that gap solitons of the same type may also exist in a nonlinear optical system consisting of focusing and defocusing tunnel-coupled planar lightguides.

Stability of mKdV?KdV Waves

We consider a combination of modified Korteweg-de Vries waves and ordinary Kortewegde Vries waves, known as mKdV-KdV waves. The Infeld-Rowlands method is developed to study the instability of mKdV-KdV waves in the limit of long wavelength perturbations.

Two-dimensional instabilities of generalized Korteweg-de Vries waves

Instabilities of the generalized Korteweg-de Vries (( u t + α 1 u m u x + α 2 u n u x + u xxx ) x + α 3 u yy = 0) waves wi th respect to two-dimensional infinitesimal longitudinal disturbances are investigated using the Infeld-Rowlands method. A linear dispersion relation expressed as a cubic equation in g̃w 1 is derived and instabilities of waves are discussed.

Stability of Modified Korteweg?de Vries Waves

A nonlinear wave theory is developed on the basis of the Infeld-Rowlands method to study the stability of modified Korteweg-de Vries waves. A general stability criterion is derived in order to show that unstable waves exist.

Modulations of KDV wavetrains

Abstract This lecture summarizes some work on the modulational theory of Korteweg de Vries wave trains which was done by H. Flaschka, G. Forest, and D.W. McLaughlin during the past year.