1,297 research outputs found
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
A new integrable generalization of the Korteweg - de Vries equation
A new integrable sixth-order nonlinear wave equation is discovered by means
of the Painleve analysis, which is equivalent to the Korteweg - de Vries
equation with a source. A Lax representation and a Backlund self-transformation
are found of the new equation, and its travelling wave solutions and
generalized symmetries are studied.Comment: 13 pages, 2 figure
Numerical integration of coupled Korteweg-de Vries System
We introduce a numerical method for general coupled Korteweg-de Vries
systems. The scheme is valid for solving Cauchy problems for arbitrary number
of equations with arbitrary constant coefficients. The numerical scheme takes
its legality by proving its stability and convergence which gives the
conditions and the appropriate choice of the grid sizes. The method is applied
to Hirota-Satsuma (HS) system and compared with its known explicit solution
investigating the influence of initial conditions and grid sizes on accuracy.
We also illustrate the method to show the effects of constants with a
transition to non-integrable cases.Comment: 11 pages, 13 figure
Collisions of acoustic solitons and their electric fields in plasmas at critical compositions
Acoustic solitons obtained through a reductive perturbation scheme are
normally governed by a Korteweg-de Vries (KdV) equation. In multispecies
plasmas at critical compositions the coefficient of the quadratic nonlinearity
vanishes. Extending the analytic treatment then leads to a modified KdV (mKdV)
equation, which is characterized by a cubic nonlinearity and is even in the
electrostatic potential. The mKdV equation admits solitons having opposite
electrostatic polarities, in contrast to KdV solitons which can only be of one
polarity at a time. A Hirota formalism has been used to derive the two-soliton
solution. That solution covers not only the interaction of same-polarity
solitons but also the collision of compressive and rarefactive solitons. For
the visualisation of the solutions, the focus is on the details of the
interaction region. A novel and detailed discussion is included of typical
electric field signatures that are often observed in ionospheric and
magnetospheric plasmas. It is argued that these signatures can be attributed to
solitons and their interactions. As such, they have received little attention.Comment: 15 pages, 15 figure
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