475 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
On asymptotically equivalent shallow water wave equations
The integrable 3rd-order Korteweg-de Vries (KdV) equation emerges uniquely at
linear order in the asymptotic expansion for unidirectional shallow water
waves. However, at quadratic order, this asymptotic expansion produces an
entire {\it family} of shallow water wave equations that are asymptotically
equivalent to each other, under a group of nonlinear, nonlocal, normal-form
transformations introduced by Kodama in combination with the application of the
Helmholtz-operator. These Kodama-Helmholtz transformations are used to present
connections between shallow water waves, the integrable 5th-order Korteweg-de
Vries equation, and a generalization of the Camassa-Holm (CH) equation that
contains an additional integrable case. The dispersion relation of the full
water wave problem and any equation in this family agree to 5th order. The
travelling wave solutions of the CH equation are shown to agree to 5th order
with the exact solution
Darboux transformation for classical acoustic spectral problem
We study discrete isospectral symmetries for the classical acoustic spectral
problem in spatial dimensions one and two, by developing a Darboux (Moutard)
transformation formalism for this problem. The procedure follows the steps,
similar to those for the Schr\"{o}dinger operator. However, there is no
one-to-one correspondence between the two problems. The technique developed
enables one to construct new families of integrable potentials for the acoustic
problem, in addition to those already known.
The acoustic problem produces a non-linear Harry Dym PDE. Using the
technique, we reproduce a pair of simple soliton solutions of this equation.
These solutions are further used to construct a new positon solution for this
PDE. Furthermore, using the dressing chain approach, we build a modified Harry
Dym equation together with its LA-pair.
As an application, we construct some singular and non-singular integrable
potentials (dielectric permitivity) for the Maxwell equations in a 2D
inhomogeneous medium.Comment: 16 pages; keywords Darboux (Moutard) transformation, Classical
acoustic spectral problem, Reflexionless potentials, Soliton
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