Two-dimensional Korteweg-de Vries Equation Research Articles

Symmetry Solutions and Conserved Vectors of the Two-Dimensional Korteweg-de Vries Equation

Symmetry Solutions and Conserved Vectors of the Two-Dimensional Korteweg-de Vries Equation

Interaction and generation of long-crested internal solitary waves in the South China Sea

[1] Oblique nonlinear interactions based on the Kadomtsev-Petviashvili equation (also known as the two-dimensional Korteweg–de Vries equation) are extended to internal solitary waves (ISWs) to explain why the amplitude does not decrease owing to the geometric spreading of the cylindrical wavefronts in the South China Sea (SCS). This resonance theory is used to explain a satellite image exhibiting special features, and it is proposed that wave arcs of different amplitude resonate, providing a mechanism for reinforcing a wave by boosting the amplitude. The present theory suggests the amplitude of the ISW that propagates across the SCS basin depends on the interaction of ISWs originating from different sources; hence, studying the generation of an ISW from a single source location cannot predict the ISW correctly.

Two-dimensional leaps in near-critical flow over isolated orography

This paper describes steady two-dimensional disturbances forced on the interface of a two-layer fluid by flow over an isolated obstacle. The oncoming flow speed is close to the linear longwave speed and the layer densities, layer depths and obstacle height are chosen so that the equations of motion reduce to the forced two-dimensional Korteweg–de Vries equation with cubic nonlinearity, i.e. the forced extended Kadomtsev–Petviashvili equation. The distinctive feature noted here is the appearance in the far lee-wave wake behind obstacles in subcritical flow of a ‘table-top’ wave extending almost one-dimensionally for many obstacles widths across the flow. Numerical integrations show that the most important parameter determining whether this wave appears is the departure from criticality, with the wave appearing in slightly subcritical flows but being destroyed by two-dimensional effects behind even quite long ridges in even moderately subcritical flow. The wave appears after the flow has passed through a transition from subcritical to supercritical over the obstacle and its leading and trailing edges resemble dissipationless leaps standing in supercritical flow. Two-dimensional steady supercritical flows are related to one-dimensional unsteady flows with time in the unsteady flow associated with a slow cross-stream variable in the two-dimensional flows. Thus the wide cross-stream extent of the table-top wave appears to derive from the combination of its occurrence in a supercritical region embedded in the subcritical flow and the propagation without change of form of table-top waves in one-dimensional unsteady flow. The table-top wave here is associated with a resonant steepening of the transition above the obstacle and a consequent twelve-fold increase in drag. Remarkably, the table-top wave is generated equally strongly and extends laterally equally as far behind an axisymmetric obstacle as behind a ridge and so leads to subcritical flows differing significantly from linear predictions.

The anomalous diffusion of wave disturbances in hydrodynamic-type systems

It is shown that solutions of many model non-linear equations of the hydrodynamic type (including the Navier-Stokes equations in the theory of a viscous fluid) in the form of localized wave disturbances undergo anomalous decay and the diffusion-type spreading under the action of Gaussian additive force noise and multiplicative parametric noise that depend randomly on time. The universality of this anomaly (i.e. its independence of the specific form of the equations and, in many respects, of the characteristic properties of the noise) is demonstrated; this reflects the generalized Galilean invariance of the hydrodynamic-type equations and the Gaussian property of random sources. The integrability of certain important hydrodynamic models enables the general conclusions to be supported by particular analytic results. Examples of expressions for the mean flow velocities, developed under the action of random forces on algebraic solitons of the completely integrable equations: equations for internal waves (the Benjamin-Ono equations), the two-dimensional Korteweg-de Vries equation (the Petviashvili-Kadomtseve equation), and the two-dimensional non-linear Schrödinger equation (the Davey-Stewartson equations), are presented in terms of the error function. It is established that for stochastic equations of the hydrodynamic type rapidly decaying multiplicative noise leads to normal Fick diffusion only when there is no additive force noise. Force noise produces anomalous diffusion of wave disturbances, which is described by Richardson's law (with a cubic relation between the temporal and squared spatial scales). The result obtained confirms the universal nature of Richardson's law, which has been demonstrated previously for many examples of turbulent and wave stochastic processes [1, 2].

A non-linear equation for drift waves

A non-linear drift wave equation is derived for a hot plasma confined by a strong magnetic field, the temperature gradient being perpendicular to the magnetic field and unidirectional. This equation, which is obtained through a long wave-length approximation, is the two-dimensional Korteweg-de Vries equation. It can be transformed approximately in the two-dimensional modified Korteweg-de Vries equation via the Miura transformation. A bi-Hamiltonian formulation of the drift wave equation is given, which implies its complete integrability. This derivation of the non-linear drift wave equation gives a strong support to k-spectrum calculations published previously by the author.

Soliton Chain Solutions to the Two-Dimensional Higher Order Korteweg-de Vries Equations of Sawada-Kotera Hierarchy

A series of two-dimensional equations with the two-dimensional Korteweg-de Vries equation being their first one are constructed as Sawada-Kotera did in one-dimensional case. Their soliton chain solutions are obtained and investigated.

A two-dimensional ion acoustic solitary wave in a weakly relativistic plasma

The two-dimensional Korteweg-de Vries equation is first derived for a weakly relativistic ion acoustic wave propagating in a collisionless plasma. We show that the relativistic effect greatly influences the phase velocity, the amplitude and the width of a solitary wave solution, and that the presence of streaming ions gives rise to the formation of a precursor. We also discuss three limiting cases of the present results.

Symmetries of the Two-Dimensional Korteweg-deVries Equation

The most general infinitesimal operator of a point symmetry for the two-dimensional Korteweg-deVries equation is given.

Explicit formulas for symmetries and conservation laws of the Kadomtsev-Petviashvili equation

Abstract Two nonlocal recursion operators are given, which yield explicit formulas for infinite hierarchies of symmetry generators and conservation laws for the two-dimensional Korteweg-de Vries equation. It is shown that the constants of the motion are in involution and that the symmetries commute.

Asymptotic behavior of the solutions of the Kadomtsev-Pyatviashvili equation (two-dimensional Korteweg-De Vries equation)

The asymptotic form of the solutions of the Kadomtsev-Pyatviashvili equation for t → ± ∞ is presented. The reverse problem of reconstructing the solution from its asymptotic form is also solved.

Interpretation of three-soliton interactions in terms of resonant triads

The three-soliton solution of the two-dimensional Korteweg-de Vries equation is analysed to show that the structure of the interaction can be represented in terms of the motion of two-soliton resonant interactions (resonant triads) as described by Miles (1977). The schematic development of the interaction with time is obtained and shown to approximate closely to computer calculations of the analytic solution. Similar results follow for interactions of more solitons and other equations.

Correspondence between the classical λø 4, double and single sine-gordon equations for three-dimensional solitons

A correspondence has been found between the classical λø 4, double and single sine-Gordon equations in 3 dimensions and multiple soliton-like solutions are given. These solutions are analogous to the resonant solutions of the two-dimensional Korteweg-de Vries equation.

The interaction ofn-dimensional soliton wave fronts

We consider solutions of the equation $$\psi $$ , where $$\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{L} = \alpha ^{ij} \partial _i \partial _j (i,j = 0,...,n)$$ and where α is any constant symmetric matrix. The solutions behave as soliton-shaped wave fronts inn dimensions. These interact but keep their identity as in one dimension. We give solutions for the interaction of up to four of these waves for any matrix α and suggest a solution forN of them. There is only one essential difference between these solutions and those of the one-dimensional system;i.e. conditions are found which restrict the motion of then-dimensional waves and it is these conditions we find the most interesting. If we choose α=diag (−1, 1, 1), then it is found that any three interacting waves intersect and form a triangle. For this α we find that the area of this triangle must remain constant in time and hence the wave fronts are forced to move in a specific fixed configuration. We also briefly consider a type of two-dimensional Korteweg-de Vries equation which, although artificially constructed from the one-dimensional case, has similar solutions to the 2-dimensional sine-Gordon equation.