113 research outputs found

    Sino-Pakistani relations in the time of COVID-19

    Get PDF
    As China's economy contracts in the wake of the Covid-19 pandemic, and with nearly $20 billion of China's Belt and Road Initiative (BRI) projects in Pakistan, policy makers in Islamabad and Beijing are keen to reaffirm the importance of their strategic partnership. Here Filippo Boni (Open University) details the recent history of Sino-Pakistan relations, and how Covid-19 has impacted them over the last few months

    NLSE on the half-line with point interactions

    Full text link
    We 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 L2L^2-subcritical or L2L^{2}-critical, while the point interaction is either attractive or repulsive. In the L2L^{2}-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 L2L^{2}-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 Impact of the China-Pakistan Economic Corridor on Pakistan’s Federal System: The Politics of the CPEC

    Get PDF
    The China-Pakistan Economic Corridor is often portrayed as the flagship project of the Belt and Road Initiative. While much attention has been devoted to its geopolitical repercussions, its impacts on Pakistan’s federal system and interprovincial relations have not yet been explored. Organized around interviews conducted in 2015, 2018, and 2019, this article demonstrates that the construction of the economic corridor is acting as a centripetal force in Pakistan’s federal structure, despite the potential for such a large external investment to redress the disparities between provinces

    Two dimensional NLS ground states with attractive Coulomb potential and point interaction

    Full text link
    We study the existence and the properties of ground states at fixed mass for a focusing nonlinear Schr\"odinger 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 L2^2-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 L2^2-critical and supercritical cases.Comment: 25 page

    Prescribed mass ground states for a doubly nonlinear Schr\"odinger equation in dimension one

    Full text link
    We investigate the problem of existence and uniqueness of ground states at fixed mass for two families of focusing nonlinear Schr\"odinger equations on the line. The first family consists of NLS with power nonlinearities concentrated at a point. For such model, we prove existence and uniqueness of ground states at every mass when the nonlinearity power is L2L^2-subcritical and at a threshold value of the mass in the L2L^2-critical regime. The second family is obtained by adding a standard power nonlinearity to the previous setting. In this case, we prove existence and uniqueness at every mass in the doubly subcritical case, namely when both the powers related to the pointwise and the standard nonlinearity are subcritical. If only one power is critical, then existence and uniqueness hold only at masses lower than the critical mass associated to the critical nonlinearity. Finally, in the doubly critical case ground states exist only at critical mass, whose value results from a non--trivial interplay between the two nonlinearities.Comment: 17 page
    corecore