2,652 research outputs found
Symbolic Computation of Conservation Laws of Nonlinear Partial Differential Equations in Multi-dimensions
A direct method for the computation of polynomial conservation laws of
polynomial systems of nonlinear partial differential equations (PDEs) in
multi-dimensions is presented. The method avoids advanced
differential-geometric tools. Instead, it is solely based on calculus,
variational calculus, and linear algebra.
Densities are constructed as linear combinations of scaling homogeneous terms
with undetermined coefficients. The variational derivative (Euler operator) is
used to compute the undetermined coefficients. The homotopy operator is used to
compute the fluxes.
The method is illustrated with nonlinear PDEs describing wave phenomena in
fluid dynamics, plasma physics, and quantum physics. For PDEs with parameters,
the method determines the conditions on the parameters so that a sequence of
conserved densities might exist. The existence of a large number of
conservation laws is a predictor for complete integrability. The method is
algorithmic, applicable to a variety of PDEs, and can be implemented in
computer algebra systems such as Mathematica, Maple, and REDUCE.Comment: To appear in: Thematic Issue on ``Mathematical Methods and Symbolic
Calculation in Chemistry and Chemical Biology'' of the International Journal
of Quantum Chemistry. Eds.: Michael Barnett and Frank Harris (2006
High-speed shear driven dynamos. Part 2. Numerical analysis
This paper aims to numerically verify the large Reynolds number asymptotic
theory of magneto-hydrodynamic (MHD) flows proposed in the companion paper
Deguchi (2019). To avoid any complexity associated with the chaotic nature of
turbulence and flow geometry, nonlinear steady solutions of the
viscous-resistive magneto-hydrodynamic equations in plane Couette flow have
been utilised. Two classes of nonlinear MHD states, which convert kinematic
energy to magnetic energy effectively, have been determined. The first class of
nonlinear states can be obtained when a small spanwise uniform magnetic field
is applied to the known hydrodynamic solution branch of the plane Couette flow.
The nonlinear states are characterised by the hydrodynamic/magnetic roll-streak
and the resonant layer at which strong vorticity and current sheets are
observed. These flow features, and the induced strong streamwise magnetic
field, are fully consistent with the vortex/Alfv\'en wave interaction theory
proposed in Deguchi (2019). When the spanwise uniform magnetic field is
switched off, the solutions become purely hydrodynamic. However, the second
class of `self-sustained shear driven dynamos' at the zero-external magnetic
field limit can be found by homotopy via the forced states subject to a
spanwise uniform current field. The discovery of the dynamo states has
motivated the corresponding large Reynolds number matched asymptotic analysis
in Deguchi (2019). Here, the reduced equations derived by the asymptotic theory
have been solved numerically. The asymptotic solution provides remarkably good
predictions for the finite Reynolds number dynamo solutions
On the completeness of solutions of Bethe's equations
We consider the Bethe equations for the isotropic spin-1/2 Heisenberg quantum
spin chain with periodic boundary conditions. We formulate a conjecture for the
number of solutions with pairwise distinct roots of these equations, in terms
of numbers of so-called singular (or exceptional) solutions. Using homotopy
continuation methods, we find all such solutions of the Bethe equations for
chains of length up to 14. The numbers of these solutions are in perfect
agreement with the conjecture. We also discuss an indirect method of finding
solutions of the Bethe equations by solving the Baxter T-Q equation. We briefly
comment on implications for thermodynamical computations based on the string
hypothesis.Comment: 17 pages; 85 tables provided as supplemental material; v2:
clarifications and references added; v3: numerical results extended to N=14,
M=
Bifurcation of Fredholm Maps I; The Index Bundle and Bifurcation
We associate to a parametrized family of nonlinear Fredholm maps
possessing a trivial branch of zeroes an {\it index of bifurcation}
which provides an algebraic measure for the number of bifurcation points from
the trivial branch. The index is derived from the index bundle of
the linearization of the family along the trivial branch by means of the
generalized -homomorphism. Using the Agranovich reduction and a
cohomological form of the Atiyah-Singer family index theorem, due to Fedosov,
we compute the bifurcation index of a multiparameter family of nonlinear
elliptic boundary value problems from the principal symbol of the linearization
along the trivial branch. In this way we obtain criteria for bifurcation of
solutions of nonlinear elliptic equations which cannot be achieved using the
classical Lyapunov-Schmidt method.Comment: 42 pages. Changes: added Lemma 2.31 and a reference + minor
corrections. To appear on TMN
Efficient computation of quasiperiodic oscillations in nonlinear systems with fast rotating parts
We present a numerical method for the investigation of quasiperiodic oscillations in applications modeled by systems of ordinary differential equations. We focus on systems with parts that have a significant rotational speed. An important element of our approach is to change coordinates into a co-rotating frame. We show that this leads to a dramatic reduction of computational effort in the case that gravitational forces can be neglected. As a practical example we study a turbocharger model for which we give a thorough comparison of results for a model with and without gravitational forces
Comparative Study of Homotopy Analysis and Renormalization Group Methods on Rayleigh and Van der Pol Equations
A comparative study of the Homotopy Analysis method and an improved
Renormalization Group method is presented in the context of the Rayleigh and
the Van der Pol equations. Efficient approximate formulae as functions of the
nonlinearity parameter for the amplitudes of the
limit cycles for both these oscillators are derived. The improvement in the
Renormalization group analysis is achieved by invoking the idea of nonlinear
time that should have significance in a nonlinear system. Good approximate
plots of limit cycles of the concerned oscillators are also presented within
this framework.Comment: 25 pages, 7 figures. Revised and upgraded: Differ Equ Dyn Syst, (26
July, 2015
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