1,171 research outputs found
Analytical Approximation Methods for the Stabilizing Solution of the Hamilton–Jacobi Equation
In this paper, two methods for approximating the stabilizing solution of the Hamilton–Jacobi equation are proposed using symplectic geometry and a Hamiltonian perturbation technique as well as stable manifold theory. The first method uses the fact that the Hamiltonian lifted system of an integrable system is also integrable and regards the corresponding Hamiltonian system of the Hamilton–Jacobi equation as an integrable Hamiltonian system with a perturbation caused by control. The second method directly approximates the stable flow of the Hamiltonian systems using a modification of stable manifold theory. Both methods provide analytical approximations of the stable Lagrangian submanifold from which the stabilizing solution is derived. Two examples illustrate the effectiveness of the methods.
Lam\'e polynomials, hyperelliptic reductions and Lam\'e band structure
The band structure of the Lam\'e equation, viewed as a one-dimensional
Schr\"odinger equation with a periodic potential, is studied. At integer values
of the degree parameter l, the dispersion relation is reduced to the l=1
dispersion relation, and a previously published l=2 dispersion relation is
shown to be partially incorrect. The Hermite-Krichever Ansatz, which expresses
Lam\'e equation solutions in terms of l=1 solutions, is the chief tool. It is
based on a projection from a genus-l hyperelliptic curve, which parametrizes
solutions, to an elliptic curve. A general formula for this covering is
derived, and is used to reduce certain hyperelliptic integrals to elliptic
ones. Degeneracies between band edges, which can occur if the Lam\'e equation
parameters take complex values, are investigated. If the Lam\'e equation is
viewed as a differential equation on an elliptic curve, a formula is
conjectured for the number of points in elliptic moduli space (elliptic curve
parameter space) at which degeneracies occur. Tables of spectral polynomials
and Lam\'e polynomials, i.e., band edge solutions, are given. A table in the
older literature is corrected.Comment: 38 pages, 1 figure; final revision
Symbolic computation of hyperbolic tangent solutions for nonlinear differential-difference equations
A new algorithm is presented to find exact traveling wave solutions of
differential-difference equations in terms of tanh functions. For systems with
parameters, the algorithm determines the conditions on the parameters so that
the equations might admit polynomial solutions in tanh.
Examples illustrate the key steps of the algorithm. Parallels are drawn
through discussion and example to the tanh-method for partial differential
equations.
The new algorithm is implemented in Mathematica. The package
DDESpecialSolutions.m can be used to automatically compute traveling wave
solutions of nonlinear polynomial differential-difference equations. Use of the
package, implementation issues, scope, and limitations of the software are
addressed.Comment: 19 pages submitted to Computer Physics Communications. The software
can be downloaded at http://www.mines.edu/fs_home/wherema
Generalized solitary and periodic wave solutions to a (2 + 1)-dimensional Zakharov-Kuznetsov equation
In this paper, the Exp-function method is employed to the Zakharov-Kuznetsov equation as a (2 + 1)-dimensional model for nonlinear Rossby waves. The observation of solitary wave solutions and periodic wave solutions constructed from the exponential function solutions reveal that our approach is very effective and convenient. The obtained results may be useful for better understanding the properties of two-dimensional coherent structures such as atmospheric blocking events. © 2009 Elsevier Inc. All rights reserved
Traveling Wave Solutions of ZK-BBM Equation Sine-Cosine Method
Travelling wave solutions are obtained by using a relatively new technique which is called sine-cosine method for ZK-BBM equations. Solution procedure and obtained results re-confirm the efficiency of the proposed scheme
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