In this Letter, we consider a coupled Schrödinger–Korteweg–de Vries equation (or Sch–KdV) equation with appropriate initial values using the Adomian’s decomposition method (or ADM). In this method, the solution is calculated in the form of a convergent power series with easily computable components. The method does not need linearization, weak nonlinearity assumptions or perturbation theory. Th...

Some types of coupled Korteweg de-Vries (KdV) equations are derived from an atmospheric dynamical system. In the derivation procedure, an unreasonable yaverage trick (which is usually adopted in literature) is removed. The derived models are classified via Painlevé test. Three types of τ -function solutions and multiple soliton solutions of the models are explicitly given by means of the exact ...

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 ...

He’s variational iteration method [1, 2], which is a modified general Lagrange multiplier method [3], has been shown to solve effectively, easily and accurately a large class of nonlinear problems with approximations which converge quickly to accurate solutions. It was successfully applied to autonomous ordinary differential equations [4], nonlinear partial differential equations with variable ...

In this paper we present an overview of the direct relation between differential invariants of projective type for curves in flat semisimple homegenous spaces and PDEs of KdV type. We describe the progress in the proof of a conjectured Theorem stating that for any such space there are geometric evolutions of curves that induce completely integrable evolutions on their invariants of projective-t...

In the past three decades, traveling wave solutions to the Korteweg–de Vries equation have been studied extensively and a large number of theoretical issues concerning the KdV equation have received considerable attention. These wave solutions when they exist can enable us to well understand the mechanism of the complicated physical phenomena and dynamical processes modeled by these nonlinear e...

Unifying hierarchies of integrable equations are discussed. They are constructed via generalized Hirota identity. It is shown that the Combescure transformations, known for a long time for the Darboux system and having a simple geometrical meaning, are in fact the symmetry transformations of generalized integrable hierarchies. Generalized equation written in terms of invariants of Combescure tr...

The standing and traveling waves are well known phenomena of the coupled ordinary differential equations in many fields. The wave solutions of the coupled system are considered to be similar to the partial differential equation of the system. In this paper, the waves which appear in a coupled magneto-elastic beam system are discussed theoretically and numerically. The physical system is continu...