Two-dimensional Korteweg Research Articles

On the solutions and conservation laws of a two-dimensional Korteweg de Vries model: Multiple exp-function method

Abstract Under investigation in this paper is a two-dimensional Korteweg de Vries model, which is a spacial extension of the Korteweg de Vries model. An infinite number of nonlocal conservation laws are given which indicate the integrability of this model. Exact soliton solutions are then respectively derived by means of the multiple exp-function method.

On the two - dimensional nonlinear Korteweg - de Vries equation with cubic stream function

In the theory of nonlinear oscillations and waves with dissipation and dispersion of the medium, it is applied the two-dimensional Korteweg - de Vries equation. In this paper a generalized nonlinear partial differential equation of third order with a cubic stream function is considered. The exact analytical solutions of two-dimensional Korteweg - de Vries equation and Kadomtsev - Petviashvili equation are obtained. The graphs effect of non-linear parameters of medium to the maximum amplitude of the waves is presented.

Analytical study on a two-dimensional Korteweg–de Vries model with bilinear representation, Bäcklund transformation and soliton solutions

With symbolic computation, Bell-polynomial scheme and bilinear method are applied to a two-dimensional Korteweg–de Vries (KdV) model, which is firstly proposed with Lax pair generating technique. Bell-polynomial expression with one auxiliary independent variable is derived and transformed into bilinear form. According to the coupled two-field conditions between the primary and replica fields, Bell-polynomial-typed Bäcklund transformations (BTs) are constructed and converted into the bilinear ones. Finally, soliton solutions of the two-dimensional KdV model are obtained (via solving the bilinear representation and BT, respectively) and compared. Such associated integrable properties as bilinear representation, BT (especially auxiliary-independent-variable-involved Bell-polynomial-typed ones constructed in this paper) and soliton solutions (especially the multi-soliton ones) may be useful for further study on other two-dimensional KdV and KdV-typed models.

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.

Soliton solutions of the two-dimensional KdV-Burgers equation by homotopy perturbation method

Abstract In this Letter, the He's homotopy perturbation method (HPM) to finding the soliton solutions of the two-dimensional Korteweg–de Vries Burgers' equation (tdKdVB) for the initial conditions was applied. Numerical solutions of the equation were obtained. The obtained solutions, in comparison with the exact solutions admit a remarkable accuracy. The results reveal that the HPM is very effective and simple.

Auto-Bäcklund transformation and new exact solutions of the generalized variable-coefficients two-dimensional Korteweg–de Vries model

Describing plasma physics, cluster physics, and geophysical fluid dynamics, a generalized variable-coefficient two-dimensional Korteweg–de Vries (GVCKdV) model is hereby under investigation. In this paper, an extended-variable-coefficient homogeneous balance method with symbolic computation is used to obtain an auto-Bäcklund transformation, some rational solutions, and new soliton-type solutions for the GVCKdV model. The solutions of the well-known cylindrical Kadomtsev-Petviashvili equation can be recovered as special cases of the results obtained here.

Travelling wave solutions of two-dimensional Korteweg–de Vries–Burgers and Kadomtsev–Petviashvili equations

The travelling wave solutions of the two-dimensional Korteweg–de Vries–Burgers and Kadomtsev–Petviashvili equations are studied from two complementary points of view. The first one is an adaptation of the factorization technique that provides particular as well as general solutions. The second one applies the Painleve analysis to both equations, throwing light on some aspects of the first method and giving an explanation to some restriction on the coefficients, as well as the relation between factorizations and integrals of motion.

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.

Two-dimensional periodic permanent waves in shallow water

When two periodic permanent wave trains on shallow water intersect obliquely, the regions of intersection are two-dimensional waves of permanent shape. This shape varies from a nearly linear superposition of the two wave trains at large angles of intersection between the wave normals, to a structure predominantly transverse to the direction of propagation at small angles of intersection. The latter shape is found to be governed by the two-dimensional Korteweg–de Vries (Kadomtsev–Petviashvili) equation. The two-dimensional permanent waves are stable to periodic disturbances parallel to their direction of propagation, but are unstable to certain oblique periodic disturbances.

Propagation of nonlinear electrostatic drift solitary waves in a collisionless plasma consisting of two ion species

A modified two-dimensional Korteweg – de Vries equation for a collisionless plasma consisting of two distinct ionic species and isothermal electrons has been derived. The effect of concentration of light ions on the amplitude and the width of the drift solitary wave is examined for an argon plasma and a helium plasma with hydrogen ion impurities. Similarly the effect of concentration of negative ions on the amplitude of the drift solitary wave is examined and it is shown that when negative ions are present one gets the usual compressive drift solitary waves with positive amplitude as well as rarefactive drift solitary waves with negative amplitude depending on the negative ion concentration.