850 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
"Dispersion management" for solitons in a Korteweg-de Vries system
The existence of ``dispersion-managed solitons'', i.e., stable pulsating
solitary-wave solutions to the nonlinear Schr\"{o}dinger equation with
periodically modulated and sign-variable dispersion is now well known in
nonlinear optics. Our purpose here is to investigate whether similar structures
exist for other well-known nonlinear wave models. Hence, here we consider as a
basic model the variable-coefficient Korteweg-de Vries equation; this has the
form of a Korteweg-de Vries equation with a periodically varying third-order
dispersion coefficient, that can take both positive and negative values. More
generally, this model may be extended to include fifth-order dispersion. Such
models may describe, for instance, periodically modulated waveguides for long
gravity-capillary waves. We develop an analytical approximation for solitary
waves in the weakly nonlinear case, from which it is possible to obtain a
reduction to a relatively simple integral equation, which is readily solved
numerically. Then, we describe some systematic direct simulations of the full
equation, which use the soliton shape produced by the integral equation as an
initial condition. These simulations reveal regions of stable and unstable
pulsating solitary waves in the corresponding parametric space. Finally, we
consider the effects of fifth-order dispersion.Comment: 19 pages, 7 figure
- …