14,250 research outputs found
Exact Travelling Wave Solutions of the Coupled Klein-Gordon Equation by the Infinite Series Method
In this paper, we employ the infinite series method for travelling wave solutions of the coupled Klein-Gordon equations. Based on the idea of the infinite series method, a simple and efficient method is proposed for obtaining exact solutions of nonlinear evolution equations. The solutions obtained include solitons and periodic solutions
Travelling waves in a nonlinear degenerate diffusion model for bacterial pattern formation
We study a reaction diffusion model recently proposed in [5] to describe the spatiotemporal evolution of the bacterium Bacillus subtilis on agar plates containing nutrient. An interesting mathematical feature of the model, which is a coupled pair of partial differential equations, is that the bacterial density satisfies a degenerate nonlinear diffusion equation. It was shown numerically that this model can exhibit quasi-one-dimensional constant speed travelling wave solutions. We present an analytic study of the existence and uniqueness problem for constant speed travelling wave solutions. We find that such solutions exist only for speeds greater than some threshold speed giving minimum speed waves which have a sharp profile. For speeds greater than this minimum speed the waves are smooth. We also characterise the dependence of the wave profile on the decay of the front of the initial perturbation in bacterial density. An investigation of the partial differential equation problem establishes,via a global existence and uniqueness argument, that these waves are the only long time solutions supported by the problem. Numerical solutions of the partial differential equation problem are presented and they confirm the results of the analysis
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 Completely Integrability Systems of Differential Equations
In this note we discuss the approach which was given by Wazwaz for the proof
of the complete integrability to the system of nonlinear differential
equations. We show that his method presented in [Wazwaz A.M. Completely
integrable coupled KdV and coupled KP systems, Commun Nonlinear Sci Simulat 15
(2010) 2828-2835] is incorrect.Comment: 14 pages. This paper was sent to the Communications in Nonlinear
Science and Numerical Simulatio
Falling liquid films with blowing and suction
Flow of a thin viscous film down a flat inclined plane becomes unstable to
long wave interfacial fluctuations when the Reynolds number based on the mean
film thickness becomes larger than a critical value (this value decreases as
the angle of inclination with the horizontal increases, and in particular
becomes zero when the plate is vertical). Control of these interfacial
instabilities is relevant to a wide range of industrial applications including
coating processes and heat or mass transfer systems. This study considers the
effect of blowing and suction through the substrate in order to construct from
first principles physically realistic models that can be used for detailed
passive and active control studies of direct relevance to possible experiments.
Two different long-wave, thin-film equations are derived to describe this
system; these include the imposed blowing/suction as well as inertia, surface
tension, gravity and viscosity. The case of spatially periodic blowing and
suction is considered in detail and the bifurcation structure of forced steady
states is explored numerically to predict that steady states cease to exist for
sufficiently large suction speeds since the film locally thins to zero
thickness giving way to dry patches on the substrate. The linear stability of
the resulting nonuniform steady states is investigated for perturbations of
arbitrary wavelengths, and any instabilities are followed into the fully
nonlinear regime using time-dependent computations. The case of small amplitude
blowing/suction is studied analytically both for steady states and their
stability. Finally, the transition between travelling waves and non-uniform
steady states is explored as the suction amplitude increases
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
Nonlinear variations in axisymmetric accretion
We subject the stationary solutions of inviscid and axially symmetric
rotational accretion to a time-dependent radial perturbation, which includes
nonlinearity to any arbitrary order. Regardless of the order of nonlinearity,
the equation of the perturbation bears a form that is similar to the metric
equation of an analogue acoustic black hole. We bring out the time dependence
of the perturbation in the form of a Li\'enard system, by requiring the
perturbation to be a standing wave under the second order of nonlinearity. We
perform a dynamical systems analysis of the Li\'enard system to reveal a saddle
point in real time, whose implication is that instabilities will develop in the
accreting system when the perturbation is extended into the nonlinear regime.
We also model the perturbation as a high-frequency travelling wave, and carry
out a Wentzel-Kramers-Brillouin analysis, treating nonlinearity iteratively as
a very feeble effect. Under this approach both the amplitude and the energy
flux of the perturbation exhibit growth, with the acoustic horizon segregating
the regions of stability and instability.Comment: 15 pages, ReVTeX. Substantially revised with respect to the previous
version. One figure and a new section on travelling waves (Sec. VI) have been
added. The bibliography has been revised. arXiv admin note: substantial text
overlap with arXiv:1207.107
Gap solitons in Bose-Einstein condensates in linear and nonlinear optical lattices
Properties of localized states on array of BEC confined to a potential,
representing superposition of linear and nonlinear optical lattices are
investigated. For a shallow lattice case the coupled mode system has been
derived. The modulational instability of nonlinear plane waves is analyzed. We
revealed new types of gap solitons and studied their stability. For the first
time a moving soliton solution has been found. Analytical predictions are
confirmed by numerical simulations of the Gross-Pitaevskii equation with
jointly acting linear and nonlinear periodic potentials.Comment: 9 pages, 14 figure
Spectral stability of nonlinear waves in KdV-type evolution equations
This paper concerns spectral stability of nonlinear waves in KdV-type
evolution equations. The relevant eigenvalue problem is defined by the
composition of an unbounded self-adjoint operator with a finite number of
negative eigenvalues and an unbounded non-invertible symplectic operator
. The instability index theorem is proven under a generic
assumption on the self-adjoint operator both in the case of solitary waves and
periodic waves. This result is reviewed in the context of other recent results
on spectral stability of nonlinear waves in KdV-type evolution equations.Comment: 15 pages, no figure
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