Korteweg-de Vries Type Research Articles

Modulational instability in nonlinear nonlocal equations of regularized long wave type

We study the stability and instability of periodic traveling waves in the vicinity of the origin in the spectral plane, for equations of Benjamin–Bona–Mahony (BBM) and regularized Boussinesq types permitting nonlocal dispersion. We extend recent results for equations of Korteweg–de Vries type and derive modulational instability indices as functions of the wave number of the underlying wave. We show that a sufficiently small, periodic traveling wave of the BBM equation is spectrally unstable to long wavelength perturbations if the wave number is greater than a critical value and a sufficiently small, periodic traveling wave of the regularized Boussinesq equation is stable to square integrable perturbations.

A Korteweg–de Vries type of fifth-order equations on a finite domain with point dissipation

The paper discusses initial value problem of a Korteweg–de Vries type of fifth-order equationwt+wxxx−wxxxxx−∑j=1najwjwx=0,w(x,0)=w0(x) posed on a periodic domain x∈[0,2π] with periodic boundary conditions wix(0,t)=wix(2π,t), i=0,2,3,4 and an L2-stabilizing feedback control law wx(2π,t)=αwx(0,t)+(1−α)wxxx(0,t) where n is a fixed positive integer, aj, j=1,2,⋯,n, α are real constants, and |α|<1. It is shown that for w0(x)∈Hα1(0,2π) with the boundary conditions described above, the problem is locally well-posed for w∈C([0,T];Hα1(0,2π)) with a conserved volume of w, [w]=∫02πw(x,t)dx. Moreover, the solution with small initial condition exists globally and approaches to [w0(x)]/(2π) as t→+∞.

The propagation of internal undular bores over variable topography

In the coastal ocean, large amplitude, horizontally propagating internal wave trains are commonly observed. These are long nonlinear waves and can be modelled by equations of the Korteweg–de Vries type. Typically they occur in regions of variable bottom topography when the variable-coefficient Korteweg–de Vries equation is an appropriate model. Of special interest is the situation when the coefficient of the quadratic nonlinear term changes sign at a certain critical point. This case has been widely studied for a solitary wave, which is extinguished at the critical point and replaced by a train of solitary waves of the opposite polarity to the incident wave, riding on a pedestal of the original polarity. Here we examine the same situation for an undular bore, represented by a modulated periodic wave train. Numerical simulations and some asymptotic analysis based on Whitham modulation equations show that the leading solitary waves in the undular bore are destroyed and replaced by a developing rarefaction wave supporting emerging solitary waves of the opposite polarity. In contrast the rear of the undular bore emerges with the same shape, but with reduced wave amplitudes, a shorter overall length scale and moves more slowly.

Об уравнениях типа Бюргерса и Кортевега-де Фриза для функций трех переменных

Об уравнениях типа Бюргерса и Кортевега-де Фриза для функций трех переменных

Nature of short, high-amplitude compressive stress pulses in a periodic dissipative laminate.

We study the evolution of high-amplitude stress pulses in periodic dissipative laminates taking into account the nonlinear constitutive equations of the components and their dissipative behavior. Aluminum-tungsten laminate was selected due to the large difference in acoustic impedances of components, the significant nonlinearity of the aluminum constitutive equation at the investigated range of stresses, and its possible practical applications. Laminates with different cell size, which controls the internal time scale, impacted by plates with different thicknesses that determine the incoming pulse duration, were investigated. It has been observed that the ratio of the duration of the incoming pulse to the internal characteristic time determines the nature of the high-amplitude dissipative propagating waves-a triangular oscillatory shock-like profile, a train of localized pulses, or a single localized pulse. These localized quasistationary waves resemble solitary waves even in the presence of dissipation: The similar pulses emerged from different initial conditions, indicating that they are inherent properties of the corresponding laminates; their characteristic length scale is determined by the scale of mesostructure, nonlinear properties of materials, and the stress amplitude; and a linear relationship exists between their speed and amplitude. They mostly recover their shapes after collision with phase shift. A theoretical description approximating the shape, length scale, and speed of these high-amplitude dissipative pulses was proposed based on the Korteweg-de Vries equation with a dispersive term determined by the mesostructure and a nonlinear term derived using Hugoniot curves of components.

Mapping Methods to Solve a Modified Korteweg-de Vries Type Equation

In this paper, we employ mapping methods to construct exact travelling wave solutions for a modified Korteweg-de Vries equation. We have derived periodic wave solutions in terms of Jacobi elliptic functions, kink solutions and singular wave solutions in terms of hyperbolic functions.

Nonlinear Wave Equations for Oceanic Internal Solitary Waves

In the coastal ocean, the interaction of barotropic tidal currents with topographic features such as the continental shelf, sills in narrow straits, and bottom ridges are often observed to generate large amplitude, horizontally propagating internal solitary waves. These are long nonlinear waves and hence can be modeled by equations of the Korteweg–de Vries type. Typically they occur in regions of variable bottom topography, with the consequence that the appropriate nonlinear evolution equation has variable coefficients. Further, as these waves can be long‐lived it is necessary to take account of the effects of the Earth's background rotation. We review this family of model evolution equations and some of their pertinent solutions, obtained both asymptotically and numerically.

Existence of travelling-wave solutions to a coupled system of Korteweg–de Vries equations

The purpose of this paper is to provide a mathematical proof of the existence of smooth solitary travelling waves for a coupled system of Korteweg–de Vries type equations {ut+uxxx+(upvp+1)x=0,vt+rvx+vxxx+(up+1vp)x=0, where u=u(x,t),v=v(x,t) are real-valued functions and r,x,t∈R. We follow a variational approach by characterizing travelling waves as minimizers of some functional under suitable constraints. Using the modern methods in the calculus of variations (the concentration-compactness principle of P. L. Lions), we prove that any minimizing sequence for the variational problem converges strongly, after an appropriate translation, to a minimizer.

Integrability Analysis of a Two-Component Burgers-Type Hierarchy

The Lax integrability of a two-component polynomial Burgers-type dynamical system is analyzed by using a differential-algebraic approach. Its linear adjoint matrix Lax representation is constructed. A related recursive operator and an infinite hierarchy of nonlinear Lax integrable dynamical systems of the Burgers–Korteweg–de-Vries type are obtained by the gradient-holonomic technique. The corresponding Lax representations are presented.

Method for Studying the Multisoliton Solutions of the Korteweg-de Vries Type Equations

We present a new approach to find travelling wave solutions for the Korteweg-de Vries type equations, which allows extending the class of known soliton solutions. Also we propose method for studying the multisoliton solutions of the Korteweg-de Vries type equations.

Global Well-Posedness for Higher-Order Schrödinger Equations in Weighted L2 Spaces

We obtain the global well-posedness for Schrödinger equations of higher orders in weighted L2 spaces. This is based on weighted L2 Strichartz estimates for the corresponding propagator with higher-order dispersion. Our method is also applied to the Airy equation which is the linear component of Korteweg-de Vries type equations.

Nonlinear Electromagnetic Waves in a Degenerate Electron-Positron Plasma

Using the reductive perturbation technique (RPT), the nonlinear propagation of magnetosonic soli- tary waves in an ultracold, degenerate (extremely dense) electron-positron (EP) plasma (containing ultracold, degen- erate electron, and positron fluids) is investigated. The set of basic equations is reduced to a Korteweg-de Vries (KdV) equation for the lowest-order perturbed magnetic field and to a KdV type equation for the higher-order perturbed mag- netic field. The solutions of these evolution equations are obtained. For better accuracy and searching on new fea- tures, the new solutions are analyzed numerically based on compact objects (white dwarf) parameters. It is found that

Nonlinear wave excitations by orbiting charged space debris objects

The nonlinear wave excitations arising from the orbital motion of a charged space debris object in the plasma environment of the Low Earth Orbital (LEO) region are investigated. In the weakly nonlinear and dispersive limit it is shown that the moving debris induced excitations can be described by a forced Korteweg de Vries (fKdV) type model equation. This equation predicts a striking phenomenon of excitation of advancing solitary waves (precursor solitons) in the upstream region of the moving object apart from weak dispersive excitations (wake fields) in the down stream region. The detection of such precursor solitons by in situ or remote means could provide a valuable diagnostic tool for detection of charged debris objects in the less than 10cm size range which are normally difficult to observe by optical means. The conditions for the excitation of such solitons through a numerical solution of the fKdV equation are delineated and the relevance and impact of these waves for the dynamical study of orbital space debris are discussed.

Poisson structure on the phase space associated to the hamiltonian dynamics of coupled Korteweg-de Vries type equations

We present the hamiltonian structures for a wide class of coupled Korteweg-de Vries systems, including the Gear and Grimshaw system that models the strong interaction of internal waves in a stratified liquid and the system of Lou, Tong, Hu and Tang that describes a two layer fluid model. Among the hamiltonian structures of these systems we found new Poisson brackets which define consistent algebras of observables.

Changing forms and sudden smooth transitions of tsunami waves

In some tsunami waves travelling over the ocean, such as the one approaching the eastern coast of Japan in 2011, the sea surface of the ocean is depressed by a small metre-scale displacement over a multi-kilometre horizontal length scale, lying in front of a positive elevation of comparable magnitude and length, which together constitute a down-up wave. Shallow water theory shows that the latter travels faster than the former, leading to an interaction, whose description is the issue addressed in this paper, using model equations of the Korteweg–de Vries type. First, we re-examine the undular bore solutions of the Korteweg–de Vries equation which describe how an initial depression wave deforms into a depression rarefaction wave followed by an undular bore of large elevation waves riding on this depression. Then we develop a new extended Korteweg–de Vries equation some of whose solutions can be used to describe the interaction of an elevation wave chasing a depression wave. These show that the two waves coincide at a given position and time producing a maximum elevation. Typically this amplitude is larger than the initial displacement magnitude by a factor which can be as large as two, which may explain anomalous elevations of tsunamis at particular positions along their trajectories. It is physically significant that for these small amplitude waves, no wave breaking occurs and there is no excess dissipation. Then, following the transition, the elevation wave moves ahead of the depression wave and the distance between them increases either linearly or logarithmically with time. The implications for how these down-up tsunami waves reach the shoreline are considered.

Hamiltonian structure for a class of parametric coupled systems of the Korteweg-de Vries type

We obtain from a lagrangian action describing a class of coupled parametric systems of KdV type its hamiltonian structure. The Poisson algebra arises from second class constraints of the theory and the use of Dirac brackets. The coupled system has relevant applications in physics.

Global well-posedness for a system of KdV-type equations with coupled quadratic nonlinearities

AbstractIn this paper, coupled systemsof Korteweg-de Vries type are considered, whereu=u(x, t),v=v(x, t) are real-valued functions and wherex, t∈R. Here, subscripts connote partial differentiation andare quadratic polynomials in the variablesuandv. Attention is given to the pure initial-value problem in whichu(x, t) andv(x, t) are both specified att= 0, namely,forx∈ ℝ. Under suitable conditions onPandQ, global well-posedness of this problem is established for initial data in theL2-based Sobolev spacesHs(ℝ) ×Hs(ℝ) for anys&amp;gt; ‒3/4.

Solutions to the modified Korteweg–de Vries equation

This is a continuation of [Notes on solutions in Wronskian form to soliton equations: Korteweg–de Vries-type, arXiv:nlin.SI/0603008]. In the present paper, we review solutions to the modified Korteweg–de Vries equation in terms of Wronskians. The Wronskian entry vector needs to satisfy a matrix differential equation set which contains complex operation. This fact makes the analysis of the modified Korteweg–de Vries to be different from the case of the Korteweg–de Vries equation. To derive complete solution expressions for the matrix differential equation set, we introduce an auxiliary matrix to deal with the complex operation. As a result, the obtained solutions to the modified Korteweg–de Vries equation are categorized into two types: solitons and breathers, together with their limit cases. Besides, we give rational solutions to the modified Korteweg–de Vries equation in Wronskian form. This is derived with the help of a Galilean transformed version of the modified Korteweg–de Vries equation. Finally, typical dynamics of the obtained solutions are analyzed and illustrated. We also list out the obtained solutions and their corresponding basic Wronskian vectors in the conclusion part.

On the existence of conservation law multiplier for partial differential equations

In this paper, we use the property of nonlinear self-adjointness with differential substitution to study the existence of conservation law multiplier for partial differential equations (PDEs). Firstly, we give a sufficient and necessary condition for the existence of the multipliers involving only independent and dependent variables, which is the nonlinear self-adjointness of the studying PDEs. Secondly, a necessary condition for the existence of the multipliers involving derivatives is given for the general evolution PDEs, which is the nonlinear self-adjointness with differential substitution. Finally, applications of multiplier and nonlinear self-adjointness with differential substitution methods to nonlinear telegraph equations and a class of Korteweg-de Vries (KdV) type equations are performed and different types of conservation laws are constructed.

On the method of strained parameters for a KdV type of equation with exact dispersion property

This paper provides an alternative methodology for analysis of three-wave interactions under the exact dispersion relation associated with gravity waves in fluid of intermediate depth. A Korteweg-de Vries type of equation with exact dispersion property is adopted as the governing equation for unidirectional wave packet evolution. Following the idea from Zakharov's seminal paper (Zakharov, V. E. (1968) Stability of periodic waves of finite amplitude on the surface of a deep fluid. \textit{Journal of Applied Mechanics and Technical Physics}, {\bf 9}, 190--194), the equation is transformed from the spatial--temporal domain to the wavenumber--temporal domain. The solution of the transformed equation is sought using the perturbation theory, for which the ansatz is expressed in the form of a regular expansion in the increasing order of a small parameter. After implementing the na\"{i}ve perturbation method, due to nonlinear mode generation and particular combinations of wavenumbers, the third-order solution contains spurious secular growth terms which appear as a consequence of resonant interaction and nonlinear mode generation. These spurious secular growth terms can be prevented by implementing the method of strained parameters for which nonlinear dispersion relation terms are produced for particular combination of wavenumbers.