On Completely Integrability Systems of Differential Equations

In this note we discuss the approach which was given by Wazwaz for the proof of the complete integrability to the system of nonlinear differential equations. We show that his method presented in [Wazwaz A.M. Completely integrable coupled KdV and coupled KP systems, Commun Nonlinear Sci Simulat 15 (2010) 2828-2835] is incorrect.Comment: 14 pages. This paper was sent to the Communications in Nonlinear Science and Numerical Simulatio

The tanh and the sine-cosine methods for the complex modified K dV and the generalized K dV equations

AbstractThe complex modified K dV (CMK dV) equation and the generalized K dV equation are investigated by using the tanh method and the sine-cosine method. A variety of exact travelling wave solutions with compact and noncompact structures are formally obtained for each equation. The study reveals the power of the two schemes where each method complements the other

Seven common errors in finding exact solutions of nonlinear differential equations

We analyze the common errors of the recent papers in which the solitary wave solutions of nonlinear differential equations are presented. Seven common errors are formulated and classified. These errors are illustrated by using multiple examples of the common errors from the recent publications. We show that many popular methods in finding of the exact solutions are equivalent each other. We demonstrate that some authors look for the solitary wave solutions of nonlinear ordinary differential equations and do not take into account the well - known general solutions of these equations. We illustrate several cases when authors present some functions for describing solutions but do not use arbitrary constants. As this fact takes place the redundant solutions of differential equations are found. A few examples of incorrect solutions by some authors are presented. Several other errors in finding the exact solutions of nonlinear differential equations are also discussed.Comment: 42 page

Exact solutions of equations for the Burgers hierarchy

Some classes of the rational, periodic and solitary wave solutions for the Burgers hierarchy are presented. The solutions for this hierarchy are obtained by using the generalized Cole - Hopf transformation

Dynamic wave solutions for (2+1)-dimensional DJKM equation in plasma physics

In this paper, we attempt to obtain exact and novel solutions for Date-Jimbo-Kashiwara-Miwa equation (DJKM) via two different techniques: Lie symmetry analysis and generalized Kudryashov method (GKM). This equation has applications in plasma physics, fluid mechanics, and other fields. The Lie symmetry method is applied to reduce the governing equation to five different ordinary differential equations (ODEs). GKM is used to obtain general and various periodic solutions. These solutions have different behaviors such as kink wave, anti-kink wave, double soliton, and single wave solution. The physical behavior of the solutions was reviewed through 2-D and 3-D graphs

On the (Non)-Integrability of KdV Hierarchy with Self-consistent Sources

Non-holonomic deformations of integrable equations of the KdV hierarchy are studied by using the expansions over the so-called "squared solutions" (squared eigenfunctions). Such deformations are equivalent to perturbed models with external (self-consistent) sources. In this regard, the KdV6 equation is viewed as a special perturbation of KdV equation. Applying expansions over the symplectic basis of squared eigenfunctions, the integrability properties of the KdV hierarchy with generic self-consistent sources are analyzed. This allows one to formulate a set of conditions on the perturbation terms that preserve the integrability. The perturbation corrections to the scattering data and to the corresponding action-angle variables are studied. The analysis shows that although many nontrivial solutions of KdV equations with generic self-consistent sources can be obtained by the Inverse Scattering Transform (IST), there are solutions that, in principle, can not be obtained via IST. Examples are considered showing the complete integrability of KdV6 with perturbations that preserve the eigenvalues time-independent. In another type of examples the soliton solutions of the perturbed equations are presented where the perturbed eigenvalue depends explicitly on time. Such equations, however in general, are not completely integrable.Comment: 16 pages, no figures, LaTe

Electrical behaviour, characteristics and properties of anodic aluminium oxide films coloured by nickel electrodeposition

Porous anodic films on 1050 aluminium substrate were coloured by AC electrodeposition of nickel. Several experiments were performed at different deposition voltages and nickel concentrations in the electrolyte in order to correlate the applied electrical power to the electrical behaviour, as well as the characteristics and properties of the coatings. The content of nickel inside the coatings reached 1.67 g/m2, depending on the experimental conditions. According to the applied AC voltage in comparison with the threshold voltage Ut, the coating either acted only as a capacitor when U\Ut and, when U[Ut, the behaviour during the anodic and cathodic parts of the power sine wave was different. In particular, due to the semi-conducting characteristics of the barrier layer, additional oxidation of the aluminium substrate occurred during the anodic part of the electrical signal, whilst metal deposition (and solvent reduction) occurred during the cathodic part; these mechanisms correspond to the blocked and pass directions of the barrier layer/electrolyte junction, respectively

Numerical study of oscillatory regimes in the Kadomtsev-Petviashvili equation

The aim of this paper is the accurate numerical study of the KP equation. In particular we are concerned with the small dispersion limit of this model, where no comprehensive analytical description exists so far. To this end we first study a similar highly oscillatory regime for asymptotically small solutions, which can be described via the Davey-Stewartson system. In a second step we investigate numerically the small dispersion limit of the KP model in the case of large amplitudes. Similarities and differences to the much better studied Korteweg-de Vries situation are discussed as well as the dependence of the limit on the additional transverse coordinate.Comment: 39 pages, 36 figures (high resolution figures at http://www.mis.mpg.de/preprints/index.html

Wavelet analysis method for solving linear and nonlinear singular boundary value problems

In this paper, a robust and accurate algorithm for solving both linear and nonlinear singular boundary value problems is proposed. We introduce the Chebyshev wavelets operational matrix of derivative and product operation matrix. Chebyshev wavelets expansions together with operational matrix of derivative are employed to solve ordinary differential equations in which, at least, one of the coefficient functions or solution function is not analytic. Several examples are included to illustrate the efficiency and accuracy of the proposed method