550 research outputs found
On elliptic solutions of the quintic complex one-dimensional Ginzburg-Landau equation
The Conte-Musette method has been modified for the search of only elliptic
solutions to systems of differential equations. A key idea of this a priory
restriction is to simplify calculations by means of the use of a few Laurent
series solutions instead of one and the use of the residue theorem. The
application of our approach to the quintic complex one-dimensional
Ginzburg-Landau equation (CGLE5) allows to find elliptic solutions in the wave
form. We also find restrictions on coefficients, which are necessary conditions
for the existence of elliptic solutions for the CGLE5. Using the investigation
of the CGLE5 as an example, we demonstrate that to find elliptic solutions the
analysis of a system of differential equations is more preferable than the
analysis of the equivalent single differential equation.Comment: LaTeX, 21 page
Solitary and periodic solutions of the generalized Kuramoto-Sivashinsky equation
The generalized Kuramoto-Sivashinsky equation in the case of the power
nonlinearity with arbitrary degree is considered. New exact solutions of this
equation are presented
On traveling waves in lattices: The case of Riccati lattices
The method of simplest equation is applied for analysis of a class of
lattices described by differential-difference equations that admit
traveling-wave solutions constructed on the basis of the solution of the
Riccati equation. We denote such lattices as Riccati lattices. We search for
Riccati lattices within two classes of lattices: generalized Lotka - Volterra
lattices and generalized Holling lattices. We show that from the class of
generalized Lotka - Volterra lattices only the Wadati lattice belongs to the
class of Riccati lattices. Opposite to this many lattices from the Holling
class are Riccati lattices. We construct exact traveling wave solutions on the
basis of the solution of Riccati equation for three members of the class of
generalized Holing lattices.Comment: 17 pages, no figure
The Nikolaevskiy equation with dispersion
The Nikolaevskiy equation was originally proposed as a model for seismic
waves and is also a model for a wide variety of systems incorporating a
neutral, Goldstone mode, including electroconvection and reaction-diffusion
systems. It is known to exhibit chaotic dynamics at the onset of pattern
formation, at least when the dispersive terms in the equation are suppressed,
as is commonly the practice in previous analyses. In this paper, the effects of
reinstating the dispersive terms are examined. It is shown that such terms can
stabilise some of the spatially periodic traveling waves; this allows us to
study the loss of stability and transition to chaos of the waves. The secondary
stability diagram (Busse balloon) for the traveling waves can be remarkably
complicated.Comment: 24 pages; accepted for publication in Phys. Rev.
A Lagrangian Description of the Higher-Order Painlev\'e Equations
We derive the Lagrangians of the higher-order Painlev\'e equations using
Jacobi's last multiplier technique. Some of these higher-order differential
equations display certain remarkable properties like passing the Painlev\'e
test and satisfy the conditions stated by Jur\'a, (Acta Appl. Math.
66 (2001) 25--39), thus allowing for a Lagrangian description.Comment: 16 pages, to be published in Applied Mathematics and Computatio
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