456 research outputs found
Seven common errors in finding exact solutions of nonlinear differential equations
We analyze the common errors of the recent papers in which the solitary wave
solutions of nonlinear differential equations are presented. Seven common
errors are formulated and classified. These errors are illustrated by using
multiple examples of the common errors from the recent publications. We show
that many popular methods in finding of the exact solutions are equivalent each
other. We demonstrate that some authors look for the solitary wave solutions of
nonlinear ordinary differential equations and do not take into account the well
- known general solutions of these equations. We illustrate several cases when
authors present some functions for describing solutions but do not use
arbitrary constants. As this fact takes place the redundant solutions of
differential equations are found. A few examples of incorrect solutions by some
authors are presented. Several other errors in finding the exact solutions of
nonlinear differential equations are also discussed.Comment: 42 page
Meromorphic solutions of nonlinear ordinary differential equations
Exact solutions of some popular nonlinear ordinary differential equations are
analyzed taking their Laurent series into account. Using the Laurent series for
solutions of nonlinear ordinary differential equations we discuss the nature of
many methods for finding exact solutions. We show that most of these methods
are conceptually identical to one another and they allow us to have only the
same solutions of nonlinear ordinary differential equations
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
Spatial chaos in weakly dispersive and viscous media: a nonperturbative theory of the driven KdV-Burgers equation
The asymptotic travelling wave solution of the KdV-Burgers equation driven by
the long scale periodic driver is constructed. The solution represents a
shock-train in which the quasi-periodic sequence of dispersive shocks or
soliton chains is interspersed by smoothly varying regions. It is shown that
the periodic solution which has the spatial driver period undergoes period
doublings as the governing parameter changes. Two types of chaotic behavior are
considered. The first type is a weak chaos, where only a small chaotic
deviation from the periodic solution occurs. The second type corresponds to the
developed chaos where the solution ``ignores'' the driver period and represents
a random sequence of uncorrelated shocks. In the case of weak chaos the shock
coordinate being repeatedly mapped over the driver period moves on a chaotic
attractor, while in the case of developed chaos it moves on a repellor. Both
solutions depend on a parameter indicating the reference shock position in the
shock-train. The structure of a one dimensional set to which this parameter
belongs is investigated. This set contains measure one intervals around the
fixed points in the case of periodic or weakly chaotic solutions and it becomes
a fractal in the case of strong chaos. The capacity dimension of this set is
calculated.Comment: 32 pages, 12 PostScript figures, useses elsart.sty and boxedeps.tex,
fig.11 is not included and can be requested from <[email protected]
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