360 research outputs found
Symbolic computation of conservation laws for nonlinear partial differential equations in multiple space dimensions
A method for symbolically computing conservation laws of nonlinear partial
differential equations (PDEs) in multiple space dimensions is presented in the
language of variational calculus and linear algebra. The steps of the method
are illustrated using the Zakharov-Kuznetsov and Kadomtsev-Petviashvili
equations as examples. The method is algorithmic and has been implemented in
Mathematica. The software package, ConservationLawsMD.m, can be used to
symbolically compute and test conservation laws for polynomial PDEs that can be
written as nonlinear evolution equations. The code ConservationLawsMD.m has
been applied to (2+1)-dimensional versions of the Sawada-Kotera, Camassa-Holm,
and Gardner equations, and the multi-dimensional Khokhlov-Zabolotskaya
equation.Comment: 26 pages. Paper will appear in Journal of Symbolic Computation
(2011). Presented at the Special Session on Geometric Flows, Moving Frames
and Integrable Systems, 2010 Spring Central Sectional Meeting of the American
Mathematical Society, Macalester College, St. Paul, Minnesota, April 10, 201
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
Complete integrability of a modified vector derivative nonlinear Schroedinger equation
Oblique propagation of magnetohydrodynamic waves in warm plasmas is described
by a modified vector derivative nonlinear Schroedinger equation, if charge
separation in Poisson's equation and the displacement current in Ampere's law
are properly taken into account. This modified equation cannot be reduced to
the standard derivative nonlinear Schroedinger equation and hence its possible
integrability and related properties need to be established afresh. Indeed, the
new equation is shown to be integrable by the existence of a bi--Hamiltonian
structure, which yields the recursion operator needed to generate an infinite
sequence of conserved densities. Some of these have been found explicitly by
symbolic computations based on the symmetry properties of the new equation.
Since the new equation includes as a special case the derivative nonlinear
Schroedinger equation, the recursion operator for the latter one is now readily
available.Comment: LateX (17 pages
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