16,319 research outputs found
New Exact Solutions of a Generalized Shallow Water Wave Equation
In this work an extended elliptic function method is proposed and applied to
the generalized shallow water wave equation. We systematically investigate to
classify new exact travelling wave solutions expressible in terms of
quasi-periodic elliptic integral function and doubly-periodic Jacobian elliptic
functions. The derived new solutions include rational, periodic, singular and
solitary wave solutions. An interesting comparison with the canonical procedure
is provided. In some cases the obtained elliptic solution has singularity at
certain region in the whole space. For such solutions we have computed the
effective region where the obtained solution is free from such a singularity.Comment: A discussion about singularity and some references are added. To
appear in Physica Script
An exactly solvable travelling wave equation in the Fisher-KPP class
For a simple one dimensional lattice version of a travelling wave equation,
we obtain an exact relation between the initial condition and the position of
the front at any later time. This exact relation takes the form of an inverse
problem: given the times at which the travelling wave reaches the
positions , one can deduce the initial profile. We show, by means of complex
analysis, that a number of known properties of travelling wave equations in the
Fisher-KPP class can be recovered, in particular Bramson's shifts of the
positions. We also recover and generalize Ebert-van Saarloos' corrections
depending on the initial condition.Comment: For version 2: some typos + clarification of (87
Periodic Travelling Waves in Dimer Granular Chains
We study bifurcations of periodic travelling waves in granular dimer chains
from the anti-continuum limit, when the mass ratio between the light and heavy
beads is zero. We show that every limiting periodic wave is uniquely continued
with respect to the mass ratio parameter and the periodic waves with the
wavelength larger than a certain critical value are spectrally stable.
Numerical computations are developed to study how this solution family is
continued to the limit of equal mass ratio between the beads, where periodic
travelling waves of granular monomer chains exist
A multiple exp-function method for nonlinear differential equations and its application
A multiple exp-function method to exact multiple wave solutions of nonlinear
partial differential equations is proposed. The method is oriented towards ease
of use and capability of computer algebra systems, and provides a direct and
systematical solution procedure which generalizes Hirota's perturbation scheme.
With help of Maple, an application of the approach to the dimensional
potential-Yu-Toda-Sasa-Fukuyama equation yields exact explicit 1-wave and
2-wave and 3-wave solutions, which include 1-soliton, 2-soliton and 3-soliton
type solutions. Two cases with specific values of the involved parameters are
plotted for each of 2-wave and 3-wave solutions.Comment: 12 pages, 16 figure
Solitary smooth hump solutions of the Camassa-Holm equation by means of the homotopy analysis method
The homotopy analysis method is used to find a family of solitary smooth hump solutions of the Camassa-Holm equation. This approximate solution, which is obtained as a series of exponentials, agrees well with the known exact solution. This paper complements the work of Wu & Liao [Wu W, Liao S. Solving solitary waves with discontinuity by means of the homotopy analysis method. Chaos, Solitons & Fractals 2005;26:177-85] who used the homotopy analysis method to find a different family of solitary wave solutions
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
Fractional Klein-Gordon equation for linear dispersive phenomena: analytical methods and applications
In this paper we discuss some exact results related to the fractional
Klein--Gordon equation involving fractional powers of the D'Alembert operator.
By means of a space-time transformation, we reduce the fractional Klein--Gordon
equation to a fractional hyper-Bessel-type equation. We find an exact analytic
solution by using the McBride theory of fractional powers of hyper-Bessel
operators. A discussion of these results within the framework of linear
dispersive wave equations is provided. We also present exact solutions of the
fractional Klein-Gordon equation in the higher dimensional cases. Finally, we
suggest a method of finding travelling wave solutions of the nonlinear
fractional Klein-Gordon equation with power law nonlinearities
On the Integrability, B\"Acklund Transformation and Symmetry Aspects of a Generalized Fisher Type Nonlinear Reaction-Diffusion Equation
The dynamics of nonlinear reaction-diffusion systems is dominated by the
onset of patterns and Fisher equation is considered to be a prototype of such
diffusive equations. Here we investigate the integrability properties of a
generalized Fisher equation in both (1+1) and (2+1) dimensions. A Painlev\'e
singularity structure analysis singles out a special case () as
integrable. More interestingly, a B\"acklund transformation is shown to give
rise to a linearizing transformation for the integrable case. A Lie symmetry
analysis again separates out the same case as the integrable one and
hence we report several physically interesting solutions via similarity
reductions. Thus we give a group theoretical interpretation for the system
under study. Explicit and numerical solutions for specific cases of
nonintegrable systems are also given. In particular, the system is found to
exhibit different types of travelling wave solutions and patterns, static
structures and localized structures. Besides the Lie symmetry analysis,
nonclassical and generalized conditional symmetry analysis are also carried
out.Comment: 30 pages, 10 figures, to appear in Int. J. Bifur. Chaos (2004
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