Coupled Korteweg De Vries Equations Research Articles

Stable and unstable vector dark solitons of coupled nonlinear Schrödinger equations: Application to two-component Bose-Einstein condensates

The dynamics of vector dark solitons in two-component Bose-Einstein condensates is studied within the framework of coupled one-dimensional nonlinear Schrödinger (NLS) equations. We consider the small-amplitude limit in which the coupled NLS equations are reduced to coupled Korteweg-de Vries (KdV) equations. For a specific choice of the parameters the obtained coupled KdV equations are exactly integrable. We find that there exist two branches of (slow and fast) dark solitons corresponding to the two branches of the sound waves. Slow solitons, corresponding to the lower branch of the acoustic wave, appear to be unstable and transform during the evolution into stable fast solitons (corresponding to the upper branch of the dispersion law). Vector dark solitons of arbitrary depths are studied numerically. It is shown that effectively different parabolic traps, to which the two components are subjected, cause an instability of the solitons, leading to a splitting of their components and subsequent decay. A simple phenomenological theory, describing the oscillations of vector dark solitons in a magnetic trap, is proposed.

Canonical structure of the coupled Korteweg–de Vries equations

The inverse problem of variational calculus is solved for the coupled Korteweg–de Vries equations resulting from a complex Lax pair. The system is found to be characterized by a second-order degenerate Lagrangian density having some common feature with the well-known Morse–Feshbach Lagrangian. The Hamiltonian structure is examined using Dirac's theory of constraints. PACS Nos.: 47.20.Ky, 42.81.Dp

Long Nonlinear Waves in Resonance with Topography

The evolution of periodic long surface waves over a periodic bottom topography resonant with the waves is studied. Coupled Korteweg–de Vries equations are derived and describe the evolution in terms of interaction between right‐ and left‐traveling waves. The coupling arises from the cumulative effect of wave scattering. We discuss the various conserved quantities of the system and compute solutions for the initial value problem and for the time‐periodic problem of fluid “sloshing” in a tank. Some three‐dimensional extensions are discussed.

Long‐Wave Instability in a Three‐Layer Stratified Shear Flow

In inviscid fluid flows, instability can occur because of a resonance between two wave modes. For the case when the modes remain distinct at the critical point where the wave phase speeds coincide, then in the weakly nonlinear, long‐wave limit, there is an expectation that the generic outcome is a model consisting of two coupled Korteweg–deVries equations. This situation is examined for a certain three‐layer stratified shear flow.

Soliton solutions in three linearly coupled Korteweg–de Vries equations

We demonstrate that a purely symmetric system of three linearly coupled Korteweg–de Vries equations, which describe the resonant interaction of internal waves in stratified fluids, possesses propagating solutions of soliton type. The amplitude equations of this system, which are derived using the multiple-scales analysis, have a similar form as the sum frequency generation equations system in diffractive medium; that have recently been the focus of much efforts. For the stationary case, the solitary wave solutions are determined in two different process of generation. In addition, we found that solutions determined in the non-stationary case are also of soliton type. A comparison is made with results obtained by more general theoretical and numerical studies reported in literature.

ON THE NONINTEGRABLE HIROTA–SATSUMA SYSTEM OF COUPLED KdV EQUATIONS IN FLUID DYNAMICS

The coupled Korteweg-de Vries (KdV) equations are important in many fields, such as nonlinear optics, fluid dynamics, supersymmetry and so on. In this paper, we modify the recently proposed variable-coefficient balancing-act algorithm, and thus obtain some exact analytic solutions for the nonintegrable Hirota–Satsuma system of coupled KdV equations, which in fluid dynamics describe interactions of two long waves with different dispersion relations.

Reduction of dispersionless coupled Korteweg–de Vries equations to the Euler–Darboux equation

A quasilinear hyperbolic system of two first-order equations is introduced. The system is linearized by means of the hodograph transformation combined with Riemann’s method of characteristics. In the process of linearization, the main step is to explicitly express the characteristic velocities in terms of the Riemann invariants. The procedure is shown to be performed by quadrature only for specific combinations of the parameters in the system. We then apply the method developed here to the dispersionless versions of the typical coupled Korteweg–de Vries (cKdV) equations including the Broer–Kaup, Ito, Hirota–Satsuma, and Bogoyavlenskii equations and show that these equations are transformed into the classical Euler–Darboux equation. A more general quasilinear system of equations is also considered with application to the dispersionless cKdV equations for the Jaulent–Miodek and Nutku–Ög̃uz equations.

A SYSTEM OF THE COUPLED KORTEWEG-DE VRIES EQUATIONS AND COMPUTERIZED SYMBOLIC COMPUTATION

In physical and engineering sciences, the studies of the Korteweg-de Vries (KdV)-typed equations are of crucial significance. Performing computerized symbolic computation, we hereby obtain an auto-Bäcklund transformation for a system of the coupled KdV equations. Sample exact analytic solutions are also presented, which are the rational solutions and solitary waves.

The Effect of Topography on the Dynamics of Interacting Solitary Waves in the Context of Atmospheric Blocking

Abstract It is shown that the influence of topography on the interaction of long, weakly nonlinear, quasigeostrophic baroclinic waves can be described by a pair of linearly coupled Korteweg–de Vries equations, with a forcing term in one of the equations. This system exhibits a rich dynamics that is suggestive of atmospheric blocking such as stable stationary solutions, transient quasi-steady-state solutions, multiple equilibria, and baroclinic instability. Topography is shown to favor the formation of blocking systems. This system is investigated both analytically, using techniques from asymptotic perturbation theory, and through numerical simulations.

The Formation of Coherent Structures in the Context of Blocking

It is shown that the interaction of long, weakly nonlinear, quasigeostrophic baroclinic waves can be described by a pair of linearly coupled Korteweg–de Vries equations. Baroclinic energy conversion is investigated as the interaction of interacting upper- and lower-layer structures, which are represented by solitary waves. This system exhibits a rich dynamics that is suggestive of atmospheric blocking features, such as stable stationary solutions, transient quasi-steady-state solutions, multiple equilibria, and baroclinic instability. This system is investigated both analytically, using techniques from asymptotic perturbation theory, and through numerical simulations.

Coupled Korteweg–de Vries equations describing, to high-order, resonant flow of a fluid over topography

The near-resonant flow of a fluid over a localized topography is examined. The flow is considered in the weakly nonlinear long-wave limit and is governed by the forced Korteweg–de Vries (fKdV) equation at first order. It is shown that the unidirectional assumption, of right-moving waves only, is incompatible with mass conservation at second order. To resolve this incompatibility, a forced coupled KdV system, which allows left-moving waves, is derived to third order (two orders beyond the fKdV approximation). The second-order fKdV equation is reformulated as an asymptotically equivalent forced Benjamin–Bona–Mahony (fBBM) equation, as its numerical scheme has superior stability. First- and second-order predictions for the resonant flow of surface water waves are compared and the mass flux between the right- and left-moving waves is found. An analytical estimate for the mass flux between the right- and left-moving waves is also derived and good agreement with the numerical solution is obtained.

A generalized Hirota–Satsuma coupled Korteweg–de Vries equation and Miura transformations

By introducing a 4×4 matrix spectral problem with three potentials, we propose a new hierarchy of nonlinear evolution equations. A typical equation in the hierarchy is a generalization of the Hirota–Satsuma coupled Korteweg–de Vries equation. Also, it is shown that the hierarchy possesses the generalized Hamiltonian form. Further, a Miura transformation related to the typical equation and its reductions are derived, from which some new coupled modified Korteweg–de Vries equations are obtained.

Painlevé classification of coupled Korteweg–de Vries systems

In this work, we give a classification of coupled Korteweg–de Vries equations. We found new systems of equations that are completely integrable in the sense of Painlevé.

Parametric envelope solitons in coupled Korteweg-de Vries equations

We demonstrate that a system of linearly coupled Korteweg-de Vries equations, which inter alia is a general model of resonantly coupled internal waves in a stratified fluid, can give rise to broad envelope solitons produced by a double phase-and group-velocity resonance between the fundamental and second harmonics for certain wavenumbers. We derive asymptotic equations for the amplitudes of the two harmonics, which are identical to the second-harmonic-generation equations in a diffractive medium, that have recently attracted a lot of attention in nonlinear optics and give rise to the so-called parametric solitons. To check if the predicted solitons are close to exact solutions of the coupled Korteweg-de Vries equations, we perform direct numerical simulations, with initial conditions suggested by the above-mentioned parametric-soliton solution to the asymptotic equations. Since the latter is known only in a numerical form, we use for them a recently developed analytical variational approximation. As a result, we observe very long-lived steadily propagating wave packets generated by these initial conditions. Thus we find a physical system that may allow experimental observation of propagating parametric solitons, while in nonlinear optics they are observed only as spatial solitons.

Decomposition of the Boussinesq equations for shallow-water waves into a set of coupled Korteweg–de Vries equations

The Boussinesq equations for shallow-water waves are decomposed into a set of equations for waves propagating in two opposite directions. Such a decomposition is valid under the assumption of slowly varying depth, and includes a nonlinear interaction term neglected in the previous derivation by Kirby and Vengayil [J. Geophys. Res. 93, 10 782 (1988)]. It is shown, however, that the decomposition is not unique, because high-order terms can be dealt with in different manners.

Solutions of an equation that lead to solutions of a modified Emden equation and a coupled Korteweg–deVries equation

In a recent paper, Moreira [Res. Rep. IF/UFRJ/83/25, Universidade Federal do Rio de Janeiro Inst. de Fisica, Cidade Univ., Ilha do Fundao, Rio de Janeiro, Brazil] obtained a nonlinear second-order differential equation that leads to the first integral of a modified Emden equation. He also obtained two particular solutions of his equation. This paper completely integrates Moreira’s equation and uses it to get a class of solutions of a coupled Korteweg–deVries (KdV) equation, recently studied by Guha Ray, Bagchi, and Sinha [J. Math. Phys. 27, 2558 (1986)].

Traveling-wave solutions and the coupled Korteweg–de Vries equation

Some coupled nonlinear equations are considered for studying traveling-wave solutions. By introducing a stream function Ψ it is shown that if one of the solutions is of the form v≡v(x−ct), the other also must be of the form u≡u(x−ct). In addition, the possibility of including cubic nonlinear terms has been considered and such a system, assuming that the solutions are of the traveling-wave type, has been solved.

A coupled Korteweg-de Vries equation with dispersion

A dispersive system describing a vector multiplet interacting with the Korteweg-de Vries field is shown to be a number of a bi-Hamiltonian integrable hierarchy.

Soliton solutions of a coupled Korteweg-de Vries equation

A coupled Korteweg-de Vries equation is presented. In general the equation exhibits a soliton solution and three basic conserved quantities. For a special choice of dispersion relations, the equation exhibits at least three-soliton solutions and two more higher-order conserved quantities.