219 research outputs found

    Lie symmetry analysis, exact solutions and conservation laws for the time fractional modified Zakharov–Kuznetsov equation

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    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

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    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

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    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

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    A weak turbulence theory is derived for magnetohydrodynamics under rapid rotation and in the presence of a large-scale magnetic field. The angular velocity Ω0\Omega_0 is assumed to be uniform and parallel to the constant Alfv\'en speed b0{\bf b_0}. Such a system exhibits left and right circularly polarized waves which can be obtained by introducing the magneto-inertial length db0/Ω0d \equiv b_0/\Omega_0. In the large-scale limit (kd0kd \to 0; kk being the wave number), the left- and right-handed waves tend respectively to the inertial and magnetostrophic waves whereas in the small-scale limit (kd+kd \to + \infty) 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 (\perp) than parallel (\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 kfk_f at which the helicity flux is injected with an inverse cascade if kfd<1k_fd < 1 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 (106\sim 10^{-6}) Rossby number is expected.Comment: 4 figures, 33 page
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