Two-dimensional Korteweg-de Vries 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

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.

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.

Symbolic Computation Study of a Generalized Variable-Coefficient Two-Dimensional Korteweg-de Vries Model with Various External-Force Terms from Shallow Water Waves, Plasma Physics, and Fluid Dynamics

Abstract The variable-coefficient two-dimensional Korteweg-de Vries (KdV) model is of considerable significance in describing many physical situations such as in canonical and cylindrical cases, and in the propagation of surface waves in large channels of varying width and depth with nonvanishing vorticity. Under investigation hereby is a generalized variable-coefficient two-dimensional KdV model with various external-force terms. With the extended bilinear method, this model is transformed into a variable-coefficient bilinear form, and then a Bäcklund transformation is constructed in bilinear form. Via symbolic computation, the associated inverse scattering scheme is simultaneously derived on the basis of the aforementioned bilinear Bäcklund transformation. Certain constraints on coefficient functions are also analyzed and finally some possible cases of the external-force terms are discussed

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.

An application of the decompositionmethod for the two-dimensional KdV-Burgers equation

A relatively new decomposition method is presented to find the explicit and numerical solutions of the two-dimensional Korteweg-de Vries Burgers' equation (tdKdVB for short) for the initial conditions. Some numerical results are also given for this equation by using the decomposition method.

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.

A Two-dimensional Boussinesq equation for water waves and some of its solutions

A two-dimensional Boussinesq equation, \[u_{tt} - u_{xx} + 3(u^2)_{xx} - u_{xxxx} - u_{yy} = 0,\] is introduced to describe the propagation of gravity waves on the surface of water, in particular the head-on collision of oblique waves. This equation combines the two-way propagation of the classical Boussinesq equation with the (weak) dependence on a second spatial variable, as occurs in the two-dimensional Korteweg-de Vries (2D KdV) (or KPII) equation. Exact and general solitary-wave, two-soliton and resonant solutions are obtained from the Hirota bilinear form of the equation. The existence of a distributed-soliton solution is investigated, but it is shown that this is not a possibility. However the connection with the classical 2D KdV equation (which does possess such a solution) is explored via a suitable parametric representation of the dispersion relation.A three-soliton solution is also constructed, but this exists only if an auxiliary constraint among the six parameters is satisfied; thus the two-dimensional Boussinesq equation is not one of the class of completely integrable equations, confirming the analysis of Hietarinta (1987). This constraint is automatically satisfied for the classical Boussinesq equation (which is completely integrable). Graphical reproductions of some of the solutions of the two-dimensional Boussinesq equations are also presented.

Korteweg-de Vries–modified Korteweg-de Vries systems and Darboux transforms in 1+1 and 2+1 dimensions

The Jaulent–Miodek hierarchy [M. Jaulent and I. Miodek, Lett. Math. Phys. 1, 243 (1976)] and its two-dimensional analog are considered from the point of view of Darboux transforms structures. A new form of those transforms as well as old fashioned forms are presented. The determinant form of solitons is obtained. The combination of different transforms in the 2+1 case allows the buildup of localized solutions in both spatial dimensions, including the two-dimensional Korteweg-de Vries (KdV) equation.

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.

On existence theorems of periodic traveling wave solutions to the two-dimensional korteweg-devries equation

The two-dimensional generalized Korteweg-deVries equation is considered. We show the existence of nonconstant periodic traveling wave solutions. The equation is converted into a nonlinear integral equation using the Green's function method. We then employ the Schauder's fixed point theorem to establish the result. An example is given for the case of f(u) = 2u 3,i.e., the modified KdV equation

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.

Traveling wave solutions to the two-dimensional Korteweg-deVries equation

Periodic wave solutions for the two-dimensional Korteweg-deVries and modified Korteweg-deVries equations are obtained. They are shown to be representable as infinite sums of solitons.

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.