Nonlinear Schrodinger equations with repulsive harmonic potential and applications

We study the Cauchy problem for Schrodinger equations with repulsive quadratic potential and power-like nonlinearity. The local problem is well-posed in the same space as that used when a confining harmonic potential is involved. For a defocusing nonlinearity, it is globally well-posed, and a scattering theory is available, with no long range effect for any superlinear nonlinearity. When the nonlinearity is focusing, we prove that choosing the harmonic potential sufficiently strong prevents blow-up in finite time. Thanks to quadratic potentials, we provide a method to anticipate, delay, or prevent wave collapse; this mechanism is explicit for critical nonlinearity.Comment: Final version, to appear in SIAM J. Math. Ana

Extension and Representation of Divergence-free Vector Fields on Bounded Domains

Let Ω ⊂ Rn be a bounded, connected domain, with b + 1 boundary components, ∂Ω = Γ0∪...∪Γb. Say O0,...,Ob are the connected components of Rn\nΩ, O0 being the unbounded component, and Γj = ∂Oj . If b > 0, pick yj ∊ Oj ; 1 ≤ j ≤ b, and set (1.1) gj(x) :=x - yj/|x - y|n, x 2 Rn \ {yj}

Isolated Eigenvalues of the Ferromagnetic Spin-J XXZ Chain with Kink Boundary Conditions

We investigate the low-lying excited states of the spin J ferromagnetic XXZ chain with Ising anisotropy Delta and kink boundary conditions. Since the third component of the total magnetization, M, is conserved, it is meaningful to study the spectrum for each fixed value of M. We prove that for J>= 3/2 the lowest excited eigenvalues are separated by a gap from the rest of the spectrum, uniformly in the length of the chain. In the thermodynamic limit, this means that there are a positive number of excitations above the ground state and below the essential spectrum

Analysis of Relaxation Time in Random Walk with Jumps

We study the relaxation time in the random walk with jumps. The random walk with jumps combines random walk based sampling with uniform node sampling and improves the performance of network analysis and learning tasks. We derive various conditions under which the relaxation time decreases with the introduction of jumps.Comment: 13 page

Existence and Stability of Standing Pulses in Neural Networks : I Existence

We consider the existence of standing pulse solutions of a neural network integro-differential equation. These pulses are bistable with the zero state and may be an analogue for short term memory in the brain. The network consists of a single-layer of neurons synaptically connected by lateral inhibition. Our work extends the classic Amari result by considering a non-saturating gain function. We consider a specific connectivity function where the existence conditions for single-pulses can be reduced to the solution of an algebraic system. In addition to the two localized pulse solutions found by Amari, we find that three or more pulses can coexist. We also show the existence of nonconvex ``dimpled'' pulses and double pulses. We map out the pulse shapes and maximum firing rates for different connection weights and gain functions.Comment: 31 pages, 29 figures, submitted to SIAM Journal on Applied Dynamical System