155 research outputs found
Infinitely many solutions for a gauged nonlinear Schrödinger equation with a perturbation
In this paper, we use the Fountain theorem under the Cerami condition to study the gauged nonlinear Schrödinger equation with a perturbation in R2. Under some appropriate conditions, we obtain the existence of infinitely many high energy solutions for the equation
Invariant measures for the periodic derivative nonlinear Schr\"odinger equation
We construct invariant measures associated to the integrals of motion of the
periodic derivative nonlinear Schr\"odinger equation (DNLS) for small data in
and we show these measures to be absolutely continuous with respect to
the Gaussian measure. The key ingredient of the proof is the analysis of the
gauge group of transformations associated to DNLS. As an intermediate step for
our main result, we prove quasi-invariance with respect to the gauge maps of
Gaussian measures on .Comment: a new result has been included, namely quasi-invariance of Gaussian
measures w.r.t. the gauge group (theorem 1.4
Invariant measures for the periodic derivative nonlinear Schrödinger equation
We construct invariant measures associated to the integrals of motion of the periodic derivative nonlinear Schrödinger equation (DNLS) for small data in L2 and we show these measures to be absolutely continuous with respect to the Gaussian measure. The key ingredient of the proof is the analysis of the gauge group of transformations associated to DNLS. As an intermediate step for our main result, we prove quasi-invariance with respect to the gauge maps of the Gaussian measure on L2 with covariance (I+(−Δ)k)−1 for any k⩾2
Ground state sign-changing solutions for critical Choquard equations with steep well potential
In this paper, we study sign-changing solution of the Choquard type equation −∆u + (λV(x) + 1) u = Iα ∗ |u| 2 |u| 2 α−2u + µ|u| p−2u in R N, where N ≥ 3, α ∈ ((N − 4) +, N), Iα is a Riesz potential, p ∈ 2 2N N−2 , 2∗ := N+α N−2 is the upper critical exponent in terms of the Hardy–Littlewood–Sobolev inequality, µ > 0, λ > 0, V ∈ C(RN, R) is nonnegative and has a potential well. By combining the variational methods and sign-changing Nehari manifold, we prove the existence and some properties of ground state sign-changing solution for λ, µ large enough. Further, we verify the asymptotic behaviour of ground state sign-changing solutions as λ → +∞ and µ → +∞, respectivel
Stability Analysis of Superconducting Electroweak Vortices
We carry out a detailed stability analysis of the superconducting vortex
solutions in the Weinberg-Salam theory described in Nucl.Phys. B826 (2010) 174.
These vortices are characterized by constant electric current and electric
charge density , for they reduce to Z strings. We consider the
generic field fluctuations around the vortex and apply the functional Jacobi
criterion to detect the negative modes in the fluctuation operator spectrum. We
find such modes and determine their dispersion relation, they turn out to be of
two different types, according to their spatial behavior. There are
non-periodic in space negative modes, which can contribute to the instability
of infinitely long vortices, but they can be eliminated by imposing the
periodic boundary conditions along the vortex. There are also periodic negative
modes, but their wavelength is always larger than a certain minimal value, so
that they cannot be accommodated by the short vortex segments. However, even
for the latter there remains one negative mode responsible for the homogeneous
expansion instability. This mode may probably be eliminated when the vortex
segment is bent into a loop. This suggests that small vortex loops balanced
against contraction by the centrifugal force could perhaps be stable.Comment: 42 pages, 11 figure
Harmonic functions and gravity localization
In models with extra dimensions, matter particles can be easily localized to
a 'brane world', but gravitational attraction tends to spread out in the extra
dimensions unless they are small. Strong warping gradients can help localize
gravity closer to the brane. In this note we give a mathematically rigorous
proof that the internal wave-function of the massless graviton is constant as
an eigenfunction of the weighted Laplacian, and hence is a power of the warping
as a bound state in an analogue Schr\"odinger potential. This holds even in
presence of singularities induced by thin branes. We also reassess the status
of AdS vacuum solutions where the graviton is massive. We prove a bound on
scale separation for such models, as an application of our recent results on KK
masses. We also use them to estimate the scale at which gravity is localized,
without having to compute the spectrum explicitly. For example, we point out
that localization can be obtained at least up to the cosmological scale in
string/M-theory solutions with infinite-volume Riemann surfaces; and in a known
class of N = 4 models, when the number of NS5- and D5-branes is roughly equal.Comment: 43 pages, 2 figure
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