Optical fibers and related partial differential equations
: IMI Short-term Joint Research Project
| Date |
| August 20-24, 2012 |
| Place |
| Meeting Room (#122), Faculty of Mathematics building, Ito Campus |
Access，Ito Campus map
| Program |
| August 20 (Monday) |
Title：Effect of basic production terms in some reaction-diffusion systems
Author：Kanako Suzuki (Ibaraki University)
We consider an activator-inhibitor system with nontrivial source terms which represent the respective basic production rates of the activator and the inhibitor. This talk is devoted to studying the effect of basic production terms on the dynamics of the system and on the shape of stationary solutions in the case where the diffusion coefficient of the inhibitor becomes infinity.
Title：A spectral theory of linear operators on Gelfand triplets and its applications to infinite dimensional dynamical systems
Author：Hayato Chiba (Kyusyu University)
Collective synchronization phenomena are well observed in a variety of areas such as engineering, biology and chemical reactions. The Kuramoto model, which is a system of differential equations on the N-torus, is often used to investigate such synchronization phenomena. In this talk, an infinite dimensional Kuramoto model is considered, and the Kuramoto's conjecture on a bifurcation diagram of the system, which is open since 1985, will be proved.
It is well known that the spectrum (eigenvalues) of a linear operator determines a local dynamics of a system of differential equations. Unfortunately, the infinite dimensional Kuramoto model has the continuous spectrum on the imaginary axis, so that the usual spectral theory does not say anything about the dynamics. To handle such continuous spectra, a new spectral theory of linear operators based on Gelfand triplets is developed. Basic notions in the usual spectral theory, such as eigenspaces, algebraic multiplicities, point/continuous/residual spectra, Riesz projections are extended to those defined on a Gelfand triplet. They prove to have the same properties as those of the usual spectral theory.
The results are applied to the Kuramoto model to prove the Kuramoto's conjecture. A center manifold theorem will be given with the aid of the Gelfand triplet and the generalized spectrum. Even if there exists the continuous spectrum on the imaginary axis, it is proved that there exists a finite dimensional center manifold on a certain space of distributions. This determines a bifurcation diagram of the Kuramoto model.
August 21 (Tuesday)
Title：Global behavior of solutions to a nonlinear equation originating in optical fibers
Author：Satoshi Masaki (Gakushuin University)
We consider power-type nonlinear Schroedinger equation, which is a generalization of model equations describing propagation of waves in optical fibers. Our interest is in time global behavior of solutions in mass-subcritical setting. We give a sharp sufficient condition for scattering phenomenon.
Title：Nonlinear Schrodinger equation describing EDFA phenomena and blow-up of the solution
Author：Naoyasu Kita (University of Miyazaki)
Recently, the special technology of light amplification has been put into practice, which is realized due to doping erbium ions into optical fibers. This technology is making use of the quantum mechanical effect called “Erbium Doped Fiber Amplification (EDFA)”. The Schrodinger equation with a complex coefficient in the nonlinearity is a mathematical model describing the propagation of optical signal through a fiber. In this talk, we observe, in mathematical way, the blowing-up result of a solution even though its initial data is sufficiently small at t = 0.
Title：Bifurcation analysis for the Lugiato-Lefever equation on a disk
Author：Tomoyuki Miyaji (Kyoto University)
Lugiato-Lefever equation is a kind of cubic nonlinear Schrodinger equation with damping and driving force. It is a model equation for pattern formation in an optical cavity. According to numerical simulations, there is a parameter region in which the system has a spatially localized and stable stationary solution. In addition, in two space dimensions, the stationary solution loses its stability as parameter changes, and a spatially localized and temporary periodic solution arises. The periodic solution disappears by a global bifurcation. In this talk, we study the equation on a disk, and we analyze the steady state mode interaction between two radially symmetric modes. We show that there can be a global bifurcation of time-periodic solution.
August 22 (Wednesday)
Title：Rigorous numerics of Saddle-saddle connections
Author：Kaname Matsue (Tohoku University)
I shall talk about a new idea of validations of saddle-saddle connections between equilibria (or invariant sets in general). Connecting orbits correspond to spatial pattern transitions like traveling waves, for example, and hence their analysis is very important. They are also important from the viewpoint of dynamical systems because they are structurally unstable and cause global bifurcations. In our method, stable and unstable manifold of equilibria can be easily described. The extension of our method to infinite dimensional dynamical systems are expected because of its simpleness.
Title：Stability of discrete breathers in nonlinear Klein-Gordon type lattices
Author：Kazuyuki Yoshimura (NTT Communication Science Laboratories)
We consider spatially localized periodic solutions called discrete breathers in one-dimensional nonlinear Klein-Gordon type lattices with weak couplings. In the limit of vanishing couplings, the system has a large number of trivial periodic solutions such that some particles oscillate periodically and the others are at rest. Existence of the discrete breathers is proved for weak couplings by continuation of the trivial periodic solutions. We have proved a theorem which determines the linear stability of the discrete breathers: the theorem shows that the stability or instability of a discrete breather depends on the phase difference and distance between the two sites in each pair of adjacent excited sites in the trivial periodic solution.
Title：Introduction to Rigorous numerics
Author：Kaname Matsue (Tohoku University)
We briefly introduce an idea of rigorous numerics using interval arithmetic. Fundamental mathematical theory with computer assistance can be applied to various problems which are difficult to solve only by pure mathematical theories. Some applications of rigorous numerics to differential equations and dynamical systems are also shown in this talk.
August 23 (Thursday)
Title：Asymptotic stability of solitary waves in the Benney-Luke model of water waves
Author：Tetsu Mizumachi (Kyushu University)
The $1$-d Benney-Luke equation is a long wave model which describes two-way water wave propagation. For this equation, as for the full water wave problem, the classic method for proving orbital stability of solitary waves fails due to the fact that solitary waves are infinitely indefinite critical points of the energy-momentum functional. In this talk, we will show asymptotic stability of solitary waves based on propagation estimates of the $1$-d Benney-Luke equation. This is a joint work with R.~L.~Pego and J.~R.~Quintero.
Title：Derivation of Bose-Hubbard model -Approximation by DNLS
Author：Reika Fukuizumi (Tohoku University)
It is known that the nonlinear Schrodinger equation with a periodic potential is approximated by a discrete nonlinear Schrodinger equation using the Wannier theory. However, this method requires some technical assumptions on the band functions, and applies only to some special periodic potentials. We thus justify this approximation using the semiclassical technique, which allows us to include more general, realistic potentials for physical applications, e. g, Bose-Einstein condensation or optic fibers. Using this approximation, we will see that for a large strength of the nonlinearity the stationary solutions turn to be localized on a single lattice site of the periodic potential. This is a joint work with A. Sacchetti.
Title：Introduction to numerical simulation by spectral methods
Author：Tomoyuki Miyaji (Kyoto University)
Several numerical methods for solving partial differential equations are known. The idea of spectral methods is to represent the solution of the equation as a sum of basis functions and to determine the coefficients to satisfy the equation. While the finite element methods use functions with compact support as the basis functions, the spectral methods use smooth orthogonal functions satisfying the boundary conditions. Although its applicability depends on the boundary conditions, if it is applicable, spectral methods provides with a highly accurate result. In this talk, we introduce numerical simulation by spectral methods.
August 24 (Friday)
Title：On weak interaction of a ground state with a nontrapping potential
Author：Masaya Maeda (Tohoku University)
We consider a nonlinear Schr\"odinger equation with nontrapping potential. We show that the ground state moving at high speed is asymptotically stable. This extends the asymptotic stability results by Cuccagna and Bambusi in the case there is no potential. This is a joint work with S. Cuccagna.