Variable-coefficient Korteweg-de Vries Equation Research Articles

Symmetries and solutions of a variable coefficient KdV equation via gauge transformation

Using a gauge transformation connecting the Korteweg–de Vries (KdV) equation and a variable coefficient KdV equation, we obtain a strong symmetry, two sets of symmetries, bilinear form and solutions for the latter from the counterparts of the KdV equation. The two sets of symmetries form a Lie algebra.

On the calculation of the timing shifts in the variable-coefficient Korteweg-de Vries equation

The variable-coefficient Korteweg-de Vries equation that governs the dynamics of weakly nonlinear long waves in a periodically variable dispersion management media is considered. For general bit patterns, an analytic expression describing the evolution of the timing shift produced by nonlinear interactions between neighboring solitons is derived. The general result with a Gaussian like solitary wave profile for a special case of negligible average dispersion is tested. By considering a piecewise-constant map, we found that the timing shift is reduced substantially in the dispersion-managed Korteweg-de Vries system.

N-SOLITON SOLUTIONS, AUTO-BÄCKLUND TRANSFORMATIONS AND LAX PAIR FOR A NONISOSPECTRAL AND VARIABLE-COEFFICIENT KORTEWEG-DE VRIES EQUATION VIA SYMBOLIC COMPUTATION

In this paper, a nonisospectral and variable-coefficient Korteweg-de Vries equation is investigated based on the ideas of the variable-coefficient balancing-act method and Hirota method. Via symbolic computation, we obtain the analytic N-soliton solutions, variable-coefficient bilinear form, auto-Bäcklund transformations (in both the bilinear form and Lax pair form), Lax pair and nonlinear superposition formula for such an equation in explicit form. Moreover, some figures are plotted to analyze the effects of the variable coefficients on the stabilities and propagation characteristics of the solitonic waves.

Wronskian Form of N-Solitonic Solution for a Variable–Coefficient Korteweg-de Vries Equation with Nonuniformities

By the symbolic computation and Hirota method, the bilinear form and an auto-Bäcklund transformation for a variable-coefficient Korteweg-de Vries equation with nonuniformities are given. Then, the N-solitonic solution in terms of Wronskian form is obtained and verified. In addition, it is shown that the (N – 1)- and N-solitonic solutions satisfy the auto-Bäcklund transformation through the Wronskian technique.

N-Soliton Solution in Wronskian Form for a Generalized Variable-Coefficient Korteweg–de Vries Equation

We concentrate on finding exact solutions for a generalized variable-coefficient Korteweg–de Vries equation of physically significance. The analytic N-soliton solution in Wronskian form for such a model is postulated and verified by direct substituting the solution into the bilinear form by virtue of the Wronskian technique. Additionally, the bilinear auto-Bäcklund transformation between the (N — 1)- and N-soliton solutions is verified.

Existence of Formal Conservation Laws of a Variable-Coefficient Korteweg–de Vries Equation from Fluid Dynamics and Plasma Physics via Symbolic Computation

Employing the method which can be used to demonstrate the infinite conservation laws for the standard Korteweg–de Vries (KdV) equation, we prove that the variable-coefficient KdV equation under the Painlevé test condition also possesses the formal conservation laws.

Evolution of solitary waves and undular bores in shallow-water flows over a gradual slope with bottom friction

This paper considers the propagation of shallow-water solitary and nonlinear periodic waves over a gradual slope with bottom friction in the framework of a variable-coefficient Korteweg–de Vries equation. We use the Whitham averaging method, using a recent development of this theory for perturbed integrable equations. This general approach enables us not only to improve known results on the adiabatic evolution of isolated solitary waves and periodic wave trains in the presence of variable topography and bottom friction, modelled by the Chezy law, but also, importantly, to study the effects of these factors on the propagation of undular bores, which are essentially unsteady in the system under consideration. In particular, it is shown that the combined action of variable topography and bottom friction generally imposes certain global restrictions on the undular bore propagation so that the evolution of the leading solitary wave can be substantially different from that of an isolated solitary wave with the same initial amplitude. This non-local effect is due to nonlinear wave interactions within the undular bore and can lead to an additional solitary wave amplitude growth, which cannot be predicted in the framework of the traditional adiabatic approach to the propagation of solitary waves in slowly varying media.

N-Solitonic Solution in Terms of Wronskian Determinant for a Perturbed Variable-Coefficient Korteweg-de Vries Equation

With the help of symbolic computation, we first derive the bilinear form and an auto-Backlund transformation for a perturbed variable-coefficient Korteweg-de Vries equation in this paper. We also construct the N-solitonic solution in the Wronskian form and give the corresponding proof via the Wronskian technique. Furthermore, the authors verify that the (N−1)- and N-solitonic solutions indeed satisfy the auto-Backlund transformation.

Symbolic computation on integrable properties of a variable-coefficient Korteweg–de Vries equation from arterial mechanics and Bose–Einstein condensates

Applicable in arterial mechanics, Bose gases of impenetrable bosons and Bose–Einstein condensates, a variable-coefficient Korteweg–de Vries (vcKdV) equation is investigated in this paper with symbolic computation. Based on the Ablowitz–Kaup–Newell–Segur system, the Lax pair and auto-Bäcklund transformation are constructed. Furthermore, the nonlinear superposition formula and an infinite number of conservation laws for the vcKdV equation are also derived. Special attention is paid to the analytic one- and two-solitonic solutions with their physical properties and possible applications discussed.

A new variable coefficient Korteweg-de Vries equation-based sub-equation method and its application to the (3 + 1)-dimensional potential-YTSF equation

With the aid of Maple, we present variable coefficient Korteweg-de Vries equation-based sub-equation method. The key idea of our method is to take advantage of the variable coefficient Korteweg-de Vries equation and its various solutions to generate various solutions of nonlinear evolution equations. The efficiency of the method can be demonstrated on the (3 + 1)-dimensional potential-YTSF equation and we construct successfully its new styles of solutions.

Energy transfer in a dispersion-managed Korteweg-de Vries system

We consider the propagation of weakly nonlinear, weakly dispersive waves in an inhomogeneous media within the framework of the variable-coefficient Korteweg-de Vries equation. An analytical formula with which to compute the energy transfer between neighboring solitary waves is derived. The resulting expression shows that the energy change in a variable KdV system is essentially due the two-wave mixing, contrary to the energy change in a nonlinear Schrödinger system, which results from the intrachannel four-wave mixing. By considering the case of Gaussian solitary wave solutions, we have determined the transfer of energy in the system analytically and numerically.

Evolution of Dipole-Type Blocking Life Cycles: Analytical Diagnoses and Observations

Abstract A variable coefficient Korteweg–de Vries (VCKdV) system is derived by considering the time-dependent background flow and boundary conditions from a nonlinear, inviscid, nondissipative, and equivalent barotropic vorticity equation in a beta plane. One analytical solution obtained from the VCKdV equation can be successfully used to explain the evolution of atmospheric dipole-type blocking (DB) life cycles. Analytical diagnoses show that the background mean westerlies have great influence on evolution of DB during its life cycle. A weak westerly is necessary for blocking development and the blocking life period shortens, accompanied with the enhanced westerlies. The shear of the background westerlies also plays an important role in the evolution of blocking. The cyclonic shear is preferable for the development of blocking but when the cyclonic shear increases, the intensity of blocking decreases and the life period of DB becomes shorter. Weak anticyclonic shear below a critical threshold is also favorable for DB formation. Time-dependent background westerly (TDW) in the life cycle of DB has some modulations on the blocking life period and intensity due to the behavior of the mean westerlies. Statistical analysis regarding the climatological features of observed DB is also investigated. Observational results show that the Pacific is a preferred region for DB, especially at high latitudes. These features may associate with the weakest westerlies and the particular westerly shear structure over the northwestern Pacific.

Soliton dynamics in a strong periodic field: The Korteweg–de Vries framework

Nonlinear long wave propagation in a medium with periodic parameters is considered in the framework of a variable-coefficient Korteweg–de Vries equation. The characteristic period of the variable medium is varied from slow to rapid, and its amplitude is also varied. For the case of a piece-wise constant coefficient with a large scale for each constant piece, explicit results for the damping of a soliton damping are obtained. These theoretical results are confirmed by numerical simulations of the variable-coefficient Korteweg–de Vries equation for the same piece-wise constant coefficient, as well as for a sinusoidally-varying coefficient. The resonance curve for soliton damping is predicted, and the maximum damping is for a soliton whose characteristic timescale is of the same order as the coefficient inhomogeneity scale. If the variation of the nonlinear coefficient is very large, and includes a critical point where the nonlinear coefficient equals to zero, the soliton breaks and is quickly damped.

Effects of Topography and Potential Vorticity Forcing on Solitary Rossby Waves in Zonally Varying Flow

A theory is developed that describes the effects of topography and potential vorticity (PV) forcing on the dynamics of solitary Rossby waves (SRWs) in zonally varying background flow. The cornerstone of the theory is the background flow, which is systematically derived rather than simply being specified as in previous theories. The evolution of the disturbance field is governed by a forced, variable coefficient Korteweg-deVries equation, which possesses SRW, low-frequency wave packet, and multiple equilibrium solutions. Although we refer to the zonally varying portion of the background flow as the stationary wave, it could just as easily be referred to as the zonally varying jet or some other feature indicating zonal variations in the background flow. The phrase “stationary” does not mean that the wave is independent of time, only that it is nonpropagating. The topography and PV forcing structures affect the evolution of the disturbance in fundamentally different ways. For example, if the topography is meridionally symmetric, it does not affect the translation speed of a meridionally antisymmetric disturbance. If the PV forcing is sufficiently weak or meridionally symmetric, it does not affect the local growth/decay of the disturbance. Conservation equations for mass, momentum, and energy are derived. For locally parallel background flow, the mass and momentum equations show that the background meridional wind and local topographic slope enter as source/sink terms. In the mass equation these terms can be combined and written solely in terms of the PV forcing. Additionally, weak or meridionally symmetric PV forcing allows the evolution equation to be written in terms of a Hamiltonian, which conserves both mass and energy. Numerical solutions of the amplitude equation reveal a richness of behaviors not found in previous studies of SRWs. These behaviors include transmission, reflection, and trapping, which may combine to produce regional confinement and local spawning of new disturbances. The regional confinement and spawning behavior only occurs when the zonally varying background flow and topography are considered in combination and displaced with respect to each other, a configuration reminiscent of observed flows in the atmosphere.

The Classical Problem of Water Waves: a Reservoir of Integrable and Nearly-Integrable Equations

In this contribution, we describe the simplest, classical problem in water waves, and use this as a vehicle to outline the techniques that we adopt to analyse this particular approach to the derivation of soliton-type equations. The surprise, perhaps, is that such an apparently transparent set of equations (the Euler equation for an incompressible fluid, the equation of mass conservation and the simplest bottom and surface conditions) contains so may different – and important – equations. We will briefly show how such equations are generated, by carefully describing the asymptotic procedures that we adopt. We will then present a number of examples, some of which will be the familiar integrable equations (Korteweg-de Vries (KdV), Nonlinear Schrödinger (NLS)) and also variants of these which are, in a sense, nearly-integrable. These will include, for example, a variable coefficient KdV equation, a nearly concentric KdV equation and higher-order NLS equation. Some of these results have significant consequences for water-wave propagation; for example, the existence of a KdV equation for arbitrary velocity profiles below the surface helps to explain why solitary waves are observed in nature. We conclude with a discussion of some of the more recent, exciting work on the rôle of the Camassa–Holm (CH) equation in water waves. We will also outline how, for this type of problem, a number of interesting and relevant variants of the CH equation can arise.

Generation of undular bores in the shelves of slowly-varying solitary waves.

We study the long-time evolution of the trailing shelves that form behind solitary waves moving through an inhomogeneous medium, within the framework of the variable-coefficient Korteweg-de Vries equation. We show that the nonlinear evolution of the shelf leads typically to the generation of an undular bore and an expansion fan, which form apart but start to overlap and nonlinearly interact after a certain time interval. The interaction zone expands with time and asymptotically as time goes to infinity occupies the whole perturbed region. Its oscillatory structure strongly depends on the sign of the inhomogeneity gradient of the variable background medium. We describe the nonlinear evolution of the shelves in terms of exact solutions to the KdV-Whitham equations with natural boundary conditions for the Riemann invariants. These analytic solutions, in particular, describe the generation of small "secondary" solitary waves in the trailing shelves, a process observed earlier in various numerical simulations. (c) 2002 American Institute of Physics.

Dispersion management for solitons in a Korteweg-de Vries system.

The existence of "dispersion-managed solitons," i.e., stable pulsating solitary-wave solutions to the nonlinear Schrodinger equation with periodically modulated and sign-variable dispersion is now well known in nonlinear optics. Our purpose here is to investigate whether similar structures exist for other well-known nonlinear wave models. Hence, here we consider as a basic model the variable-coefficient Korteweg-de Vries equation; this has the form of a Korteweg-de Vries equation with a periodically varying third-order dispersion coefficient, that can take both positive and negative values. More generally, this model may be extended to include fifth-order dispersion. Such models may describe, for instance, periodically modulated waveguides for long gravity-capillary waves. We develop an analytical approximation for solitary waves in the weakly nonlinear case, from which it is possible to obtain a reduction to a relatively simple integral equation, which is readily solved numerically. Then, we describe some systematic direct simulations of the full equation, which use the soliton shape produced by the integral equation as an initial condition. These simulations reveal regions of stable and unstable pulsating solitary waves in the corresponding parametric space. Finally, we consider the effects of fifth-order dispersion. (c) 2002 American Institute of Physics.

POSITONIC SOLUTIONS OF A VARIABLE-COEFFICIENT KdV EQUATION IN FLUID DYNAMICS AND PLASMA PHYSICS

Positons are a new concept in nonlinear sciences. In this paper, we report some positonic, exact analytic solutions for a variable-coefficient Korteweg-de Vries (KdV) equation arising from fluid dynamics and plasma physics.

Hamiltonian formulation for solitary waves propagating on a variable background

Solitary waves propagating on a variable background are conventionally described by the variable-coefficient Korteweg-de Vries equation. However, the underlying physical system is often Hamiltonian, with a conserved energy functional. Recent studies for water waves and interfacial waves have shown that an alternative approach to deriving an appropriate evolution equation, which asymptotically approximates the Hamiltonian, leads to an alternative variable-coefficient Korteweg-de Vries equation, which conserves the underlying Hamiltonian structure more explicitly. This paper examines the relationship between these two evolution equations, which are asymptotically equivalent, by first discussing the conservation laws for each equation, and then constructing asymptotically a slowly-varying solitary wave.

Hamiltonian formulation for the description of interfacial solitary waves

Abstract. We consider solitary waves propagating on the interface between two fluids, each of constant density, for the case when the upper fluid is bounded above by a rigid horizontal plane, but the lower fluid has a variable depth. It is well-known that in this situation, the solitary waves can be described by a variable-coefficient Korteweg-de Vries equation. Here we reconsider the derivation of this equation and present a formulation which preserves the Hamiltonian structure of the underlying system. The result is a new variable-coefficient Korteweg-de Vries equation, which conserves energy to a higher order than the more conventional well-known equation. The new equation is used to describe the transformation of an interfacial solitary wave which propagates into a region of decreasing depth.