1,721,019 research outputs found
Two dimensional NLS ground states with attractive Coulomb potential and point interaction
Full text linkWe study the existence and the properties of ground states at fixed mass for a focusing nonlinear Schroedinger equation in dimension two with a point interaction, an attractive Coulomb potential and a nonlinearity of power type. We prove that for any negative value of the Coulomb charge, for any positive value of the mass and for any L2-subcritical power nonlinearity, such ground states exist and exhibit a logarithmic singularity where the interaction is placed. Moreover, up to multiplication by a phase factor, they are positive, radially symmetric and decreasing. An analogous result is obtained also for minimizers of the action restricted to the Nehari manifold, getting the existence also in the L2-critical and supercritical cases
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The China-Pakistan Economic Corridor at Ten: Taking Stock of the Belt and Road’s “Flagship” Project
No full textThis chapter presents some of the underlying dynamics characterizing the evolution of the China-Pakistan Economic Corridor (CPEC), the Belt and Road Initiative's (BRI's) “flagship project.” We introduce three interweaving themes that are then developed in greater depth in the chapters included in this volume: what the case of CPEC means for Chinese global agency; the domestic political transformations in Pakistan that were set in motion by Chinese investments; and how the resulting projects have played out on the ground. In doing so, we suggest that the BRI is best understood as an interactive process between China and its international partners, which is producing interdependent relations between them. Contrary to prevalent narratives about the BRI and CPEC specifically, there is little evidence of a rising China simply reshaping the world to its liking and imposing its strategic designs on subservient countries. Rather, BRI projects are the product of complex negotiations, a source of unexpected setbacks and frustrations, and exercise a significant adaptive pressure back on China itself
NLS Ground States on a Hybrid Plane
Full text linkWe study the existence, the nonexistence, and the shape of the ground states of a Nonlinear Schrödinger Equation on a manifold called hybrid plane, that consists of a half-line whose origin is connected to a plane. The nonlinearity is of power type, focusing and subcritical. The energy is the sum of the Nonlinear Schrödinger energies with a contact interaction on the half-line and on the plane with an additional quadratic term that couples the two components. By ground state we mean every minimizer of the energy at a fixed mass. As a first result, we single out the following rule: a ground state exists if and only if the confinement near the junction is energetically more convenient than escaping at infinity along the halfline, while escaping through the plane is shown to be never convenient. The problem of existence reduces then to a competition with the one-dimensional solitons. By this criterion, we prove existence of ground states for large and small values of the mass. Moreover, we show that at given mass a ground state exists if one of the following conditions is satisfied: the interaction at the origin of the half-line is not too repulsive; the interaction at the origin of the plane is sufficiently attractive; the coupling between the half-line and the plane is strong enough. On the other hand, nonexistence holds if the contact interactions on the half-line and on the plane are repulsive enough and the coupling is not too strong. Finally, we provide qualitative features of ground states. In particular, we show that in the presence of coupling every ground state is supported both on the half-line and on the plane and each component has the shape of a ground state at its mass for the related Nonlinear Schrödinger energy with a suitable contact interaction. These are the first results for the Nonlinear Schrödinger Equation on a manifold of mixed dimensionality
NLS ground states on the half-line with point interactions
Full text linkWe investigate the existence and the uniqueness of NLS ground states of fixed
mass on the half-line in the presence of a point interaction at the origin. The
nonlinearity is of power type, and the regime is either -subcritical or
-critical, while the point interaction is either attractive or
repulsive. In the -subcritical case, we prove that ground states exist
for every mass value if the interaction is attractive, while ground states
exist only for sufficiently large masses if the interaction is repulsive. In
the latter case, if the power is less or equal to four, ground states coincide
with the only bound state. If instead, the power is greater than four, then
there are values of the mass for which two bound states exist, and neither of
the two is a ground state, and values of the mass for which two bound states
exist, and one of them is a ground state. In the -critical case, we
prove that ground states exist for masses strictly below a critical mass value
in the attractive case, while ground states never exist in the repulsive case.Comment: 17 page
The search for NLS ground states on a hybrid domain: Motivations, methods, and results
Full text linkWe discuss the problem of establishing the existence of the Ground States for the subcritical focusing Nonlinear Schrödinger energy on a domain made of a line and a plane intersecting at a point. The problem is physically motivated by the experimental realization of hybrid traps for Bose-Einstein Condensates, that are able to concentrate the system on structures close to the domain we consider. In fact, such a domain approximates the trap as the temperature approaches the absolute zero. The spirit of the paper is mainly pedagogical, so we focus on the formulation of the problem and on the explanation of the result, giving references for the technical points and for the proofs
An explicitly solvable NLS model with discontinuous standing waves
No full textWe study the NLS Equation on the line with a point interaction given by the superposition of an attractive delta potential with a dipole interaction, in the cases of L2-subcritical and L2-critical nonlinearity. For a subcritical nonlinearity we prove the existence and the uniqueness of Ground States at any mass. If the mass exceeds an explicit threshold, then there exists a positive excited state too. For the critical nonlinearity we prove that Ground States exist only in a specific interval of masses, while in a different interval excited states exist. We provide the value of the optimal constant in the Gagliardo-Nirenberg estimate and describe in the dipole case the branches of the stationary states as the strength of the interaction varies. Since all stationary states are explicitly computed, ours is a solvable model involving a non-standard interplay of a nonlinearity with a point interaction
Competing nonlinearities in NLS equations as source of threshold phenomena on star graphs
Full text linkExistence, structure, and robustness of ground states of a NLSE in 3D with a point defect
Full text linkWe study the ground states for the Schr\"odinger equation with a focusing
nonlinearity and a point interaction in dimension three. We establish that
ground states exist for every value of the mass; moreover they are positive,
radially symmetric, decreasing along the radial direction, and show a
Coulombian singularity at the location of the point interaction. Remarkably,
the existence of the ground states is independent of the attractive or
repulsive character of the point interaction.Comment: 17 pages. Keywords: standing waves, nonlinear Schr\"odinger, ground
states, delta interaction, radially symmetric solutions, rearrangements.
Accepted for publication in the Special Collection on the Proceedings of ICMP
XX and in a regular issue of Journal of Mathematical Physics. arXiv admin
note: text overlap with arXiv:2109.0948
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