4,296 research outputs found
Hidden solitons in the Zabusky-Kruskal experiment: Analysis using the periodic, inverse scattering transform
Recent numerical work on the Zabusky--Kruskal experiment has revealed,
amongst other things, the existence of hidden solitons in the wave profile.
Here, using Osborne's nonlinear Fourier analysis, which is based on the
periodic, inverse scattering transform, the hidden soliton hypothesis is
corroborated, and the \emph{exact} number of solitons, their amplitudes and
their reference level is computed. Other "less nonlinear" oscillation modes,
which are not solitons, are also found to have nontrivial energy contributions
over certain ranges of the dispersion parameter. In addition, the reference
level is found to be a non-monotone function of the dispersion parameter.
Finally, in the case of large dispersion, we show that the one-term nonlinear
Fourier series yields a very accurate approximate solution in terms of Jacobian
elliptic functions.Comment: 10 pages, 4 figures (9 images); v2: minor revision, version accepted
for publication in Math. Comput. Simula
Closed geodesics and billiards on quadrics related to elliptic KdV solutions
We consider algebraic geometrical properties of the integrable billiard on a
quadric Q with elastic impacts along another quadric confocal to Q. These
properties are in sharp contrast with those of the ellipsoidal Birkhoff
billiards. Namely, generic complex invariant manifolds are not Abelian
varieties, and the billiard map is no more algebraic. A Poncelet-like theorem
for such system is known. We give explicit sufficient conditions both for
closed geodesics and periodic billiard orbits on Q and discuss their relation
with the elliptic KdV solutions and elliptic Calogero systemComment: 23 pages, Latex, 1 figure Postscrip
Stability of Attractive Bose-Einstein Condensates in a Periodic Potential
Using a standing light wave trap, a stable quasi-one-dimensional attractive
dilute-gas Bose-Einstein condensate can be realized. In a mean-field
approximation, this phenomenon is modeled by the cubic nonlinear Schr\"odinger
equation with attractive nonlinearity and an elliptic function potential of
which a standing light wave is a special case. New families of stationary
solutions are presented. Some of these solutions have neither an analog in the
linear Schr\"odinger equation nor in the integrable nonlinear Schr\"odinger
equation. Their stability is examined using analytic and numerical methods.
Trivial-phase solutions are experimentally stable provided they have nodes and
their density is localized in the troughs of the potential. Stable
time-periodic solutions are also examined.Comment: 12 pages, 18 figure
Elliptic equations with transmission and Wentzell boundary conditions and an application to steady water waves in the presence of wind
In this paper, we present results about the existence and uniqueness of
solutions of elliptic equations with transmission and Wentzell boundary
conditions. We provide Schauder estimates and existence results in H\"older
spaces. As an application, we develop an existence theory for small-amplitude
two-dimensional traveling waves in an air-water system with surface tension.
The water region is assumed to be irrotational and of finite depth, and we
permit a general distribution of vorticity in the atmosphere.Comment: 33 page
Periodic solutions for critical fractional problems
We deal with the existence of -periodic solutions to the following
non-local critical problem \begin{equation*} \left\{\begin{array}{ll}
[(-\Delta_{x}+m^{2})^{s}-m^{2s}]u=W(x)|u|^{2^{*}_{s}-2}u+ f(x, u) &\mbox{in}
(-\pi,\pi)^{N} \\ u(x+2\pi e_{i})=u(x) &\mbox{for all} x \in \mathbb{R}^{N},
\quad i=1, \dots, N, \end{array} \right. \end{equation*} where , , , is the fractional critical
Sobolev exponent, is a positive continuous function, and is a
superlinear -periodic (in ) continuous function with subcritical
growth. When , the existence of a nonconstant periodic solution is
obtained by applying the Linking Theorem, after transforming the above
non-local problem into a degenerate elliptic problem in the half-cylinder
, with a nonlinear Neumann boundary
condition, through a suitable variant of the extension method in periodic
setting. We also consider the case by using a careful procedure of limit.
As far as we know, all these results are new.Comment: Calculus of Variations and Partial Differential Equations (2018
Dirichlet and Neumann Problems for String Equation, Poncelet Problem and Pell-Abel Equation
We consider conditions for uniqueness of the solution of the Dirichlet or the
Neumann problem for 2-dimensional wave equation inside of bi-quadratic
algebraic curve. We show that the solution is non-trivial if and only if
corresponding Poncelet problem for two conics associated with the curve has
periodic trajectory and if and only if corresponding Pell-Abel equation has a
solution.Comment: Published in SIGMA (Symmetry, Integrability and Geometry: Methods and
Applications) at http://www.emis.de/journals/SIGMA
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