Coupled Korteweg De Vries Equations Research Articles

Energy preserving integration of bi-Hamiltonian partial differential equations

The energy preserving average vector field (AVF) integrator is applied to evolutionary partial differential equations (PDEs) in bi-Hamiltonian form with nonconstant Poisson structures. Numerical results for the Korteweg de Vries (KdV) equation and for the Ito type coupled KdV equation confirm the long term preservation of the Hamiltonians and Casimir integrals, which is essential in simulating waves and solitons. Dispersive properties of the AVF integrator are investigated for the linearized equations to examine the nonlinear dynamics after discretization.

Soliton propagation and collision in a variable-coefficient coupled Korteweg-de Vries equation

In this paper, a variable-coefficient coupled Korteweg-de Vries equation is presented and studied by Hirota bilinear method. The multi-soliton solutions expressed in the form of Pfaffians are obtained. We further analyze dynamic characters of these soliton solutions. The appearances of resonant soliton behaviors involving some novel soliton fusion and fission phenomena have been reported.

Approximate symmetry reduction to the perturbed coupled KdV equations derived from two-layer fluids

Purpose: The purpose of this paper is to research the solution of the perturbed coupled Korteweg-de Vries (KdV) equations. Methods: The results are obtained through researching a system of corresponding partial differential equations which approximates the perturbed coupled KdV equations and the symmetry method. Results: The symmetry structure and the corresponding general approximate symmetry reduction are derived. Then, using the symmetry reduction, the infinite series solutions are obtained. Conclusions: The approximate symmetry reduction approach can be used to search for similar results of other perturbed nonlinear differential equations with small parameters.

Hamiltonian Structures and Integrability for a Discrete Coupled KdV-Type Equation Hierarchy

Coupled Korteweg-de Vries (KdV) systems have many important physical applications. By considering a 4 × 4 spectral problem, we derive a discrete coupled KdV-type equation hierarchy. Our hierarchy includes the coupled Volterra system proposed by Lou et al. (e-print arXiv:0711.0420) as the first member which is a discrete version of the coupled KdV equation. We also investigate the integrability in the Liouville sense and the multi-Hamiltonian structures for the obtained hierarchy.

Multisoliton, multipositon, multinegaton, and multiperiodic solutions of a coupled Volterra lattice system and their continuous limits

This paper aims to find new explicit solutions including multisoliton, multipositon, multinegaton, and multiperiodic for a coupled Volterra lattice system. This coupled lattice system is an integrable discrete version of the coupled Korteweg-deVries (KdV) equation which has many physical applications. The dynamical properties of these new solutions are discussed in detail. We also prove that the theory of the coupled Volterra lattice system including the Lax pair, the Darboux transformation, and explicit solutions yield the corresponding theory of the coupled KdV equation in the continuous limit.

Approximate analytic solutions for a generalized Hirota–Satsuma coupled KdV equation and a coupled mKdV equation

This paper applies the variational iteration method to obtain approximate analytic solutions of a generalized Hirota–Satsuma coupled Korteweg-de Vries (KdV) equation and a coupled modified Korteweg-de Vries (mKdV) equation. This method provides a sequence of functions which converges to the exact solution of the problem and is based on the use of the Lagrange multiplier for the identification of optimal values of parameters in a functional. Some examples are given to demonstrate the reliability and convenience of the method and comparisons are made with the exact solutions.

A generalized coupled Korteweg–de Vries hierarchy, bi-Hamiltonian structure, and Darboux transformation

A new generalized coupled Korteweg–de Vries (KdV) hierarchy is presented starting from a 4×4 matrix spectral problem with four potentials. Its generalized bi-Hamiltonian structure is also investigated by using the trace identity. Most importantly, a new generalized coupled KdV equation is produced. Moreover, a Darboux transformation for the generalized coupled KdV equation is established with the aid of the gauge transformation between the corresponding 4×4 matrix spectral problems, by which some explicit solutions of the generalized coupled KdV equation are obtained. As a reduction, a Darboux transformation of the KdV equation and its explicit solutions are given.

Steady gap solitons in a coupled Korteweg–de Vries system: A dynamical systems approach

In this article we search for steady gap soliton solutions of a set of coupled Korteweg–de Vries equations. These solutions arise whenever there is a narrow gap in the linear spectrum. For small amplitudes we use a dynamical systems approach combined with a normal form analysis to find a canonical equation set. When truncated at the cubic order in amplitude we can exhibit explicit solutions which agree with those found earlier by an asymptotic analysis. We argue that two of these solutions persist under perturbation by the higher-order terms.

Exact travelling wave solutions of the dissipative coupled Korteweg-de Vries equation

This paper employs the theory of planar dynamical systems and undetermined coefficient method to study travelling wave solutions of the dissipative coupled Korteweg-de Vries equation. The possible kink profile solitary wave solutions and approximate damped oscillatory solutions of the equation are obtained by using undetermined coefficient method. Error estimates indicate that the approximate solutions are meaningful.

Prolongation structures of a generalized coupled Korteweg-de Vries equation and Miura transformation

In this paper, the prolongation structures of a generalized coupled Korteweg-de Vries (KdV) equation are investigated and two integrable coupled KdV equations associated with their Lax pairs are derived. Furthermore, a Miura transformation related to a integrable coupled KdV equation is derived, from which a new coupled modified KdV equations is obtained.

ON THE CONTROLLABILITY OF A COUPLED SYSTEM OF TWO KORTEWEG–DE VRIES EQUATIONS

This paper proves the local exact boundary controllability property of a nonlinear system of two coupled Korteweg–de Vries equations which models the interactions of weakly nonlinear gravity waves (see [10]). Following the method in [24], which combines the analysis of the linearized system and the Banach's fixed point theorem, the controllability problem is reduced to prove a nonstandard unique continuation property of the eigenfunctions of the corresponding differential operator.

Darboux transformation for a generalized Hirota-Satsuma coupled Korteweg–de Vries equation

A Darboux transformation for the generalized Hirota-Satsuma coupled Korteweg-de Vries (KdV) equation is derived with the aid of the gauge transformation between the corresponding 4x4 matrix spectral problems with three potentials, by which some explicit solutions of the generalized Hirota-Satsuma coupled KdV equation are constructed. As a reduction, a Darboux transformation of the complex coupled KdV equation and its explicit solutions are obtained.

Multi soliton solution for the system of Coupled Korteweg-de Vries equations

In this paper we consider the system of coupled Korteweg-de Vries equations and establish the transformation which turns the coupled KdV equations into the single nonlinear partial differential equation, then we obtain an auto-Bäcklund transformation and Lax pairs by using the extended homogeneous balance method. Also, we obtain the multi soliton solutions and the new trigonometric function periodic solutions by using the generalized tanh method.

Numerical solution of a coupled Korteweg–de Vries equations by collocation method

AbstractA numerical method for solving the coupled Korteweg‐de Vries (CKdV) equation based on the collocation method with quintic B‐spline finite elements is set up to simulate the solution of CKdV equation. Invariants and error norms are studied wherever possible to determine the conservation properties of the algorithm. Simulation of single soliton, interaction of two solitons, and birth of solitons are presented. A linear stability analysis shows the scheme to be unconditionally stable. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2009

A direct method to find solutions of some type of coupled Korteweg–de Vries equations using hyperelliptic functions of genus two

Abstract We suggest how one can obtain exact solutions of some type of coupled Korteweg–de Vries equations by means of hyperelliptic functions of genus two.

Lie symmetry analysis and reduction of a new integrable coupled KdV system

In this paper, Lie symmetry is investigated for a new integrable coupled Korteweg–de Vries (KdV) equation system. Using some symmetry subalgebra of the equation system, we obtain five types of the significant similarity reductions. Abundant solutions of the coupled KdV equation system, such as the solitary wave solution, exponential solution, rational solution and polynomial solution, etc. are obtained from the reduced equations. Especially, one type of group-invariant solution of reduced equations can be acquired by means of the Painleve I transcendent function.

A New Coupled KdV Equation: Painlevé Test

A new type of coupled Korteweg de-Vries equation isfound to be Painlevé-integrable. The new model is a specialcase which can be used to describe two-layer fluids with differentdispersion relations.

Comment on “Stable and unstable vector dark solitons of coupled nonlinear Schrödinger equation: Application to two-component Bose-Einstein condensates

In a recent paper, V. A. Brazhnyi and V. V. Konoto [Phys. Rev. E 72, 026616 (2005)] investigated the dynamics of vector dark solitons in two-component Bose-Einstein condensates. In the small amplitude limit, they deduced a coupled Korteweg-de Vries equation from the coupled Gross-Pitaevskii equations. They found that two branches of (slow and fast) dark solitons corresponding to the two branches of the sound waves exist. The slow solitons, corresponding to the lower branch of the acoustic wave, appear to be unstable and transform during the evolution into the stable fast solitons (corresponding to the upper branch of the dispersion law). However, our discussion shows that these results are incorrect.

Coupled KdV equations derived from two-layer fluids

Some types of coupled Korteweg de-Vries (KdV) equations are derived from an atmospheric dynamical system. In the derivation procedure, an unreasonable $y$-average trick (which is usually adopted in literature) is removed. The derived models are classified via Painlev\'e test. Three types of $\tau$-function solutions and multiple soliton solutions of the models are explicitly given by means of the exact solutions of the usual KdV equation. It is also interesting that for a non-Painlev\'e integrable coupled KdV system there may be multiple soliton solutions.

Dynamics of Solitary-waves in the Coupled Korteweg-De Vries Equations

We find new analytic solitary-wave solutions, having a nonzero background at infinity, of the coupled Korteweg-De Vries equation, using the auxiliary function method. We study the dynamical properties of the solitary-waves by numerical simulations. It is shown that the solitary-waves can be stable or unstable, depending on the coefficients of the model. We study the interaction dynamics by using the solitary-waves as initial profiles to show that the mass and energy of the coupled Korteweg- De Vries can be conserved for a negative third-order dispersion term. - PACS numbers: 03.40.Kf, 02.30.Jr, 47.20.Ky, 52.35.Mw