7,431 research outputs found
A numerical method with properties of consistency in the energy domain for a class of dissipative nonlinear wave equations with applications to a Dirichlet boundary-value problem
In this work, we present a conditionally stable finite-difference scheme that
consistently approximates the solution of a general class of (3+1)-dimensional
nonlinear equations that generalizes in various ways the quantitative model
governing discrete arrays consisting of coupled harmonic oscillators.
Associated with this method, there exists a discrete scheme of energy that
consistently approximates its continuous counterpart. The method has the
properties that the associated rate of change of the discrete energy
consistently approximates its continuous counterpart, and it approximates both
a fully continuous medium and a spatially discretized system. Conditional
stability of the numerical technique is established, and applications are
provided to the existence of the process of nonlinear supratransmission in
generalized Klein-Gordon systems and the propagation of binary signals in
semi-unbounded, three-dimensional arrays of harmonic oscillators coupled
through springs and perturbed harmonically at the boundaries, where the basic
model is a modified sine-Gordon equation; our results show that a perfect
transmission is achieved via the modulation of the driving amplitude at the
boundary. Additionally, we present an example of a nonlinear system with a
forbidden band-gap which does not present supratransmission, thus establishing
that the existence of a forbidden band-gap in the linear dispersion relation of
a nonlinear system is not a sufficient condition for the system to present
supratransmission
Spatiotemporal chaotic dynamics of solitons with internal structure in the presence of finite-width inhomogeneities
We present an analytical and numerical study of the Klein-Gordon kink-soliton
dynamics in inhomogeneous media. In particular, we study an external field that
is almost constant for the whole system but that changes its sign at the center
of coordinates and a localized impurity with finite-width. The soliton solution
of the Klein-Gordon-like equations is usually treated as a structureless
point-like particle. A richer dynamics is unveiled when the extended character
of the soliton is taken into account. We show that interesting spatiotemporal
phenomena appear when the structure of the soliton interacts with finite-width
inhomogeneities. We solve an inverse problem in order to have external
perturbations which are generic and topologically equivalent to well-known
bifurcation models and such that the stability problem can be solved exactly.
We also show the different quasiperiodic and chaotic motions the soliton
undergoes as a time-dependent force pumps energy into the traslational mode of
the kink and relate these dynamics with the excitation of the shape modes of
the soliton.Comment: 10 pages Revtex style article, 22 gziped postscript figures and 5 jpg
figure
Numerical Analysis and Applications of the Process of Nonlinear Supratransmission in Mechanical Systems of Coupled Oscillators with Damping
In this paper we develop a finite-difference scheme to approximate radially symmetric solutions and (1 + 1)-dimensional solutions of the initial-value problem with smooth initial conditions Ý2w Ýt2. Þ2w. ƒÀ Ý Ýt..Þ2w + ƒÁ Ýw Ýt + m2w + GŒ(w) = 0, subject to : ( w(Px, 0) = ƒÓ(Px), Px ¸ D, Ýw Ýt (Px, 0) = ƒÕ(Px), Px ¸ D, in an open sphere D around the origin, where the internal and external damping coefficients ƒÀ and ƒÁ, respectively, are constant. The functions ƒÓ and ƒÕ are radially symmetric in D, they are small at infinity, and rƒÓ(r) and rƒÕ(r) are also assumed to be small at infinity. We prove that our scheme is consistent order O( t2) + O( r2) for GŒ identically equal to zero, and provide a necessary condition for it to be stable order n. A cornerstone of our investigation will be the study of potential applications of our model to discrete versions involving nonlinear systems of coupled oscillators. More concretely, we make use of the process of nonlinear supratransmission of energy in these chain systems and our numerical techniques in order to transmit binary information. Our simulations show that, under suitable parametric conditions, the transmission of binary signals can be achieved successfully
Numerical Analysis and Applications of the Process of Nonlinear Supratransmission in Mechanical Systems of Coupled Oscillators with Damping
In this paper we develop a finite-difference scheme to approximate radially symmetric solutions and (1 + 1)-dimensional solutions of the initial-value problem with smooth initial conditions Ý2w Ýt2. Þ2w. ƒÀ Ý Ýt..Þ2w + ƒÁ Ýw Ýt + m2w + GŒ(w) = 0, subject to : ( w(Px, 0) = ƒÓ(Px), Px ¸ D, Ýw Ýt (Px, 0) = ƒÕ(Px), Px ¸ D, in an open sphere D around the origin, where the internal and external damping coefficients ƒÀ and ƒÁ, respectively, are constant. The functions ƒÓ and ƒÕ are radially symmetric in D, they are small at infinity, and rƒÓ(r) and rƒÕ(r) are also assumed to be small at infinity. We prove that our scheme is consistent order O( t2) + O( r2) for GŒ identically equal to zero, and provide a necessary condition for it to be stable order n. A cornerstone of our investigation will be the study of potential applications of our model to discrete versions involving nonlinear systems of coupled oscillators. More concretely, we make use of the process of nonlinear supratransmission of energy in these chain systems and our numerical techniques in order to transmit binary information. Our simulations show that, under suitable parametric conditions, the transmission of binary signals can be achieved successfully
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