219 research outputs found
Lie symmetry analysis, exact solutions and conservation laws for the time fractional modified Zakharov–Kuznetsov equation
In this work, Lie symmetry analysis (LSA) for the time fractional modified Zakharov–Kuznetsov (mZK) equation with Riemann–Liouville (RL) derivative is analyzed. We transform the time fractional mZK equation to nonlinear ordinary differential equation (ODE) of fractional order using its point symmetries with a new dependent variable. In the reduced equation, the derivative is in Erdelyi–Kober (EK) sense. We obtained exact traveling wave solutions by using fractional DξαG/G-expansion method. Using Ibragimov's nonlocal conservation method to time fractional nonlinear partial differential equations (FNPDEs), we compute conservation laws (CLs) for the mZK equation
Symbolic computation of exact solutions expressible in hyperbolic and elliptic functions for nonlinear PDEs
Algorithms are presented for the tanh- and sech-methods, which lead to
closed-form solutions of nonlinear ordinary and partial differential equations
(ODEs and PDEs). New algorithms are given to find exact polynomial solutions of
ODEs and PDEs in terms of Jacobi's elliptic functions.
For systems with parameters, the algorithms determine the conditions on the
parameters so that the differential equations admit polynomial solutions in
tanh, sech, combinations thereof, Jacobi's sn or cn functions. Examples
illustrate key steps of the algorithms.
The new algorithms are implemented in Mathematica. The package
DDESpecialSolutions.m can be used to automatically compute new special
solutions of nonlinear PDEs. Use of the package, implementation issues, scope,
limitations, and future extensions of the software are addressed.
A survey is given of related algorithms and symbolic software to compute
exact solutions of nonlinear differential equations.Comment: 39 pages. Software available from Willy Hereman's home page at
http://www.mines.edu/fs_home/whereman
Invariants of Generalized Fifth Order Non-Linear Partial Differential Equation
The fifth order non-linear partial differential equation in generalized form is analyzed for Lie symmetries. The classical Lie group method is performed to derive similarity variables of this equation and the ordinary differential equations (ODEs) are deduced. These ordinary differential equations are further studied and some exact solutions are obtained
Weak turbulence theory for rotating magnetohydrodynamics and planetary dynamos
A weak turbulence theory is derived for magnetohydrodynamics under rapid
rotation and in the presence of a large-scale magnetic field. The angular
velocity is assumed to be uniform and parallel to the constant
Alfv\'en speed . Such a system exhibits left and right circularly
polarized waves which can be obtained by introducing the magneto-inertial
length . In the large-scale limit (; being
the wave number), the left- and right-handed waves tend respectively to the
inertial and magnetostrophic waves whereas in the small-scale limit () pure Alfv\'en waves are recovered. By using a complex helicity
decomposition, the asymptotic weak turbulence equations are derived which
describe the long-time behavior of weakly dispersive interacting waves {\it
via} three-wave interaction processes. It is shown that the nonlinear dynamics
is mainly anisotropic with a stronger transfer perpendicular () than
parallel () to the rotating axis. The general theory may converge to
pure weak inertial/magnetostrophic or Alfv\'en wave turbulence when the large
or small-scales limits are taken respectively. Inertial wave turbulence is
asymptotically dominated by the kinetic energy/helicity whereas the
magnetostrophic wave turbulence is dominated by the magnetic energy/helicity.
For both regimes a family of exact solutions are found for the spectra which do
not correspond necessarily to a maximal helicity state. It is shown that the
hybrid helicity exhibits a cascade whose direction may vary according to the
scale at which the helicity flux is injected with an inverse cascade if
and a direct cascade otherwise. The theory is relevant for the
magnetostrophic dynamo whose main applications are the Earth and giant planets
for which a small () Rossby number is expected.Comment: 4 figures, 33 page
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